The Reagan administration, in an effort to keep alive its proposal for a "New Federalism," has suggested letting the federal government continue to finance and run the costly food stamp program.

In his State of the Union address, President Reagan proposed turning over responsibility for the food stamp and welfare programs to the states in exchange for federal takeover of the Medicaid program.

While enthusiastic about federalization of Medicaid, state legislators and governors had strongly resisted accepting administration of food stamps and welfare, contending that these are properly the responsibility of the federal government.

The new offer, presented by top presidential aides to representatives of the nation's state legislators at a White House meeting yesterday, appeared to be an effort to win the legislators' support and an endorsement by the governors. Those are regarded as critical to any federalism bill's chances on Capitol Hill.

The stalemate over Reagan's proposed "swap" with the states of income-security programs for the poor had resulted in delays in drafting legislation and prompted news reports declaring the proposal dead.

After their meeting yesterday with White House chief of staff James A. Baker III, counselor Edwin Meese III and intergovernmental affairs assistant Richard S. Williamson, state legislative representatives said a compromise can be worked out

"They were very encouraging and wanting to rekindle the federalism initiative," said Mary Jane Gallagher, press secretary for the National Conference of State Legislatures.

While refusing to discuss the details of proposals offered in the meeting, Williamson said, "Obviously, we would not have been discussing in such a serious and particularized vein if we aren't serious about trying to work out a swap in the very near future."

The White House aides told the legislators' representatives that they want to send a bill to Congress by early May, more than a month after the deadline the White House had originally set for drafting legislation when Reagan first unveiled the federalism proposal in the State of the Union message.

Capitol Hill aides, state and local officials and White House aides have indicated privately that they do not expect a federalism bill to be passed by both houses of Congress this year but would like hearings to be held in the hope that action would come next year.

Administration aides have begun drafting legislation for a second, less controversial, component of the president's federalism proposal in which state and local governments would be given the option of ending a number of federal grant programs.

Under that component, called the "turnback," a federal trust fund would be created, and state and local governments would draw from it a sum roughly equal to the cost of the old categorical programs.

State legislative representatives said presidential aides told them yesterday that it is possible that the "turnback" proposal might be sent to the Hill first as negotiations continue over the "swap" but that they prefer to send one package.

Much of the discussion yesterday was devoted to methods for structuring a federalized Medicaid program, a subject of great concern to governors, particularly those in the big industrial states where medical benefits for the indigent are broader and more generous.

One option presented by the White House was that the federal government would pay the bills, with the states continuing to set eligibility standards. Those states desiring to be more generous would have the extra cost deducted from their share of the "turnback" trust fund.