It was word of Minnesota's generosity that made Allena Volante pack up her 6-year-old daughter and leave suburban Dallas a month ago. She said she believed she would find here what she could not in Texas: social programs that could ease her life as a struggling single mother.

"I didn't want to come," she said, "because it's too cold."

But her sister, a Minnesota resident, told her that Volante could qualify for state medical assistance here.

"Medical was the biggest reason I came," she said, describing how her asthma would send her to the emergency room and leave her with a $3,000 bill she could not begin to pay.

Although she was working as a data entry clerk in Texas, she could not get insurance through her employer and "it wasn't worth the hassle" to apply for Medicaid because she did not believe she would qualify. In fact, eligibility standards for Medicaid are more generous in Minnesota than Texas and Minnesota also offers a separate public assistance program to pay medical bills for those in need.

Studies show that families like Volante's, who move solely in pursuit of higher welfare benefits, make up a small fraction of the thousands of newcomers to Minneapolis each year. But that handful is driving a contentious debate over whether the state, with its unstinting package of social programs, has become a "welfare magnet" for the poor from states with lower benefits or tougher eligibility requirements.

That question, which has echoed for years through Minnesota politics, is resounding into a national debate, with the Senate poised to resume consideration today of legislation that would allow states much greater freedom to design their own welfare programs.

Many researchers argue that under the pending bills, the disparities among state welfare benefits will increase dramatically, and that could encourage more welfare families to uproot themselves in search of bigger checks.

That could put pressure on lawmakers, worried that their states may become welfare m agnets, to cut benefits. And that, many argue, could set off a desperate competition among the states.

"States don't want to be the ones stuck with all the poor welfare mothers," said University of Michigan demographer William Frey. While Frey does not believe that many people move today for welfare benefits alone, that could change under the proposed bills, he said.

Frey and other experts say even if the number who actually move for this reason remains relatively small, that will be less important than the fear of becoming a magnet. "What I think is a myth could get to be accepted, and that will drive policy," he said.

Like other predictions of how states will behave under a dramatic federal welfare overhaul, this scenario is hotly contested. Everyone agrees that much would change if the Republican legislation becomes law, ending the long-standing system under which eligible poor are guaranteed benefits and the federal government automatically matches state spending on welfare and Medicaid costs.

But supporters of the GOP legislation say that replacing the federal role with "block grants" -- lump-sum payments to the states, will not trigger a rapid "race to the bottom" among states.

"Each state has a history with the program . . . part and parcel of the state's culture in dealing with at-risk families," said Gerald Whitburn, secretary of the Massachusetts Executive Office of Health and Human Services. "My instinct is that they won't be turned upside down overnight under any circumstances."

Nevertheless, some states already are looking for ways to avoid drawing the out-of-state poor. Wisconsin and California received permission from the federal government to pay newcomers who apply for welfare in certain counties at the level of benefits they would have received in their previous state. The California program was struck down by a federal appeals court on grounds that it was unconstitutional.

Minnesota, where a similar, statewide law was struck down in federal court last year, is preparing to request a federal waiver to institute such a program on a limited basis. In Hennepin County, which includes Minneapolis, some welfare families who have come in from out of state are offered a free bus ticket if they want to leave, county officials said.

"There is a growing political recognition that an import policy is not acceptable," said Minnesota Gov. Arne Carlson (R). "We know it's showing up in crime, welfare, prisons. . . . We do not want to be in the importing business. We will devise a range of policies to make sure we take care of Minnesotans, but we're not in the business of subsidizing Gary, Indiana."

That fear has prompted Hennepin County to look carefully at its welfare caseload. A 1993 study found that about 40 percent of welfare applicants had moved to the county within the previous year, most from out of state. Of those, this and other studies concluded, about 7 percent may have moved because of higher welfare benefits, a figure derived by surveying new applicants and studying migration data.

In Minnesota, the maximum grant for a parent and two children was $532 a month under Aid to Families with Dependent Children (AFDC), the nation's basic cash welfare program. That compared with $367 in Illinois, the source of many of Minneapolis's newcomers. But welfare officials point out that families in states with low payments receive more in federal food stamps than those in high-benefit states, narrowing the gap.

Even those who come seeking higher benefits often cite a mixture of reasons for choosing Minnesota. Allena Volante said her prime reason for moving was to get medical coverage, but she also had a sister here to live with and a niece to baby-sit Volante's daughter, Jaquita.

Within a week of her move, Volante had found a job as an inspector at a manufacturing plant, where she said she earns $9.50 an hour. But she said she isn't eligible for health coverage through her employer for a year and even then the premiums for herself and her daughter would be prohibitive -- $260 a month.

So Volante, who has never married and receives no child support from Jaquita's father, is applying for Medicaid or, failing that, the state's medical assistance program for the working poor.

Minnesota's largess is no secret, Volante said.

"I know a lot of people who abuse the system, people I know from Chicago who come up here," she said. "I was telling my friends in Texas about the benefits up here."

But as Volante filled out application forms in the county welfare office last week, other families filling out the same forms said they moved here to find jobs, escape bad schools and violent neighborhoods or to be near their families.

Tangie Singleton, a 23-year-old single mother of four, said her AFDC payments would rise to about $700 monthly, compared with the $485 she was receiving in Chicago.

Still, she said, "I moved mostly because my mom was up here. It's nice to be around somebody." If her mother were living elsewhere, she would have gone there, she said, regardless of the welfare benefits.

Singleton paid $156 for the Greyhound bus ticket to bring her family and her boyfriend here a month ago, she said. She did not work in Chicago and in Minneapolis she spends her days searching for an apartment, she said. In the meantime, her family stays in an emergency shelter across the street from the welfare office.

Singleton, who dropped out of high school, cannot receive AFDC until she finds an apartment, but she receives another form of public assistance to pay for her family to stay in the shelter.

Despite the research evidence that welfare benefits play a marginal role, stories like Singleton's fan the volatile political controversy over how easy the county and state should make it for newcomers to receive benefits and what steps they should take to slow the influx.

Unlike most midwestern states, Minnesota's population increased in the 1980s, largely in Minneapolis and the surrounding county. According to the 1990 Census, about a tenth of Hennepin County's population arrived here from other states between 1985 and 1990.

Generally, newcomers were more likely than others to use county services, from health care to drug treatment, mental health facilities and the emergency family shelters. At the same time, the study found that the number of families on AFDC leaving Minnesota was comparable to the number arriving.

Hennepin County Commissioner Mike Opat said he is approached often and everywhere by constituents angry about their tax dollars going to pay for social programs to help outsiders. "If other states engage in a race to the bottom, Minnesota will be at some point forced to reduce the scope of its benefits as well," said Opat, a Democrat. "It's like we're being hung with our own noose. We created a very responsible and kind system to help people out." But, he said, "It's being abused."

Academic researchers tend to agree that welfare migration is small, but real.

A 1982 Michigan study concluded it was "highly probable" that 3.4 percent of the welfare applicants had been drawn to the state by the benefits. In a 1994 Wisconsin survey of newcomers who applied for welfare, 20 percent cited higher welfare benefits as important in their decision to move.

But researchers say even if the numbers remain relatively low, several changes in the pending welfare legislation could heighten the pressure on lawmakers to lower benefits: The legislation would create greater incentives to cut, they say. Now, a state that cuts welfare payments by $100 saves between $20 and $50 of its own money. The rest, which comes from federal matching funds, returns to the federal treasury.

But under the proposed legislation, a state would receive a predetermined block grant no longer linked to state spending. And so if the state chooses to cut welfare benefits by $100 under that system, it would save all $100.

Also, under law, states now are required to spend some of their own money to get a federal match. But under the House bill, states could eliminate all welfare spending and still keep the block grant. The pending Senate bill would end any state spending requirement after two years.

Finally, both the House and Senate bills limit federal welfare payments to any family to five years and allow states to lower that limit to two years. That sets up another point of competition between states, experts say. A state with a five-year limit adjacent to a state with a two-year limit will feel great pressure to prevent neighboring welfare recipients who hit their limit from moving over the border.

"The incentives to race to the bottom, or at least avoid becoming a welfare magnet, are going to at least double under the proposed arrangement," said Harvard University Prof. Paul Peterson, who co-wrote the 1990 book, "Welfare Magnets."

Those who dispute the "race to the bottom" theory say another piece of the pending legislation will be equally important: the requirement that welfare recipients move into jobs.

"Work requirements are going to be a much bigger factor" than benefit levels, said Gerald Miller, Michigan's director of social services. "Clients may say, If you're going to require me to go to this training, I may as well go get this job.' "

In Minneapolis, there is no disputing the arrival of newcomers from Chicago and elsewhere, whatever their motivations. In the emergency family shelter downtown, operated with public funds by a private group known as People Service People, officials say 90 percent of the residents have arrived from out of state. Dawn Holcomb, a 24-year-old mother of four, said she moved four months ago from Davenport, Iowa, because her aunt was here and she heard it was easier to find a job.

"I was shocked," she said, when she learned that the $495 monthly AFDC payment she received in Iowa would grow to $697 in Minnesota.

"I'd rather work anyway," she said, "and set a positive image for my kids."

Holcomb found a telemarketing job easily, but then quit, saying she needed to spend her time looking for an apartment so she could move out of the shelter. She said she intends to go back to her job, where she earned $7 an hour plus commission.

Last week, the shelter was nearly full with 927 people, about half of them children. The grassy spot across the street has become known as a hangout for shelter residents and others who have recently adopted Minnesota as their home. They bring with them a kind of folklore: stories of hand-printed signs in Chicago advertising the shelter here, and bus tickets from caseworkers in other states eager to get rid of another welfare family. CAPTION: DISPARATE BENEFITS (This graphic was not available)