The word from here on President Reagan's new budget is that state officials will live happily with less money if Reagan will cut the strings attached to it.

"If you're going to cut the budget," said Republican Gov. Christopher (Kit) Bond, "make sure you do not do it without relieving us of some of the regulatory burden."

Reagan made the federal burden on the states a target as he crisscrossed the country last year in pursuit of the presidency. Now it is his problem to solve, and as the situation here shows, there is a lot to do.

Missouri hasn't had an underground coal mine in about 40 years, but in one of the government office buildings here a state employe is dutifully working on a plan to regulate underground mining. The reason: federal regulations.

Some blocks away, the head of the state's health programs hands a visitor a 29-page document from Washington. It contains 104 recommendations, many subdivided into several items, for changing one health program in the state. It says: "Efforts should continue to fill all vacant staff Community Health Nursing positions with qualified nurses, preferably at the baccalaureate level."

"It was de Tocqueville, I think, who said the genius of the American system is that we get the kind of government we deserve," Bond said. "We've gotten more than we deserve from Washington."

But there is an inevitable clash coming over whether these federal regulations are truly burdensome, or simply requirements to force states to meet the needs of the people. Much of the federal government's role came by default, as state and local officials ignored obvious problems, and while state officials argue that Washington has carried its responsibilities too far they also acknowledge that Washington has been effective in identifying problems and providing the impetus to solve them.

Now as the flow of money from Washington slows, these state officials want to assume the power that has been taken by Washington and say they are confident they can do the job better.

Kit Bond is in his second term as governor of Missouri. He was first elected in 1972, a 32-year-old Republican whiz kid who wanted to reform the Democrats in Jefferson City. Four years later, he was out of work, upset by Democrat Joe Teasdale.

Last November, Bond recaptured the governorship in a bitter, personal fight with Teasdale. One of the differences he sees already is the size of the federal government's role in his state.

"There's been a continued growth in the federal government," Bond said in an interview here. "The cumulative effect of all the little nitpicking things, the regulations, the hassle, has become substantial."

Like other governors, Bond says he would like more flexibility to use federal money to solve Missouri's problems.

"We've got a high infant mortality rate in this state," he said. "We need to concentrate our resources on that until we get it solved. Now you get funds for 40 different programs, and in each of those you've got to spend the money for which it was specified."

Bond also recalls with some bitterness the special-education program he proposed in his first term, just before Washington came along with its own program. "We had worked out a system that we thought would serve the needs of children in Missouri, and it was overridden at the federal level," he said.

Bond is at war with the Environmental Protection Agency over how to spend sewage-treatment grants.The federal government, Bond said, has urged the state to spend the money on secondary treatment plants in St. Louis and Kansas City, the two largest cities.

Bond and Fred Lafser, director of natural resources, would prefer to spend the EPA money on secondary or tertiary treatment facilities in other cities on streams smaller than the Mississippi and Missouri rivers.

"We've got smaller streams in the state where the flow is very drastically affected by smaller communities, and these are some of our highly valued rivers and streams. I say the state ought to have flexibility in determining priorities on where to spend that money," Bond said.

President Reagan's proposal to limit future increases in Medicaid spending will fall heavily on states, which share the costs, unless some of the federal rules are eliminated, as the governors told Reagan during their winter meeting in Washington.

Last year, the cost of a daily stay increased 46 percent at one hospital in Missouri, 13 percent at another. Nonetheless, the state was required to cover the increases at both places.

"The effect of federal regulations allows inflation to be passed on to consumers, which in this case is the state," said Barrett Toan, the state's director of social services.

Toan also complained about the clause in regulations requiring freedom of choice for Medicaid users. Its purpose is to prevent a dual system of health care, one for the rich and another for the poor.

But from Toan's viewpoint, it contributes to higher costs, which already have created a $37 million deficit in state financing of the program.

Bond said he would prefer to have much of the Medicaid work done at the large public hospitals, which he said were underused, and Toan suggested that the state could negotiate bulk rates that would cut costs considerably.

"We could save 20 percent on the per diem rate," he said, which would mean a $30 million state savings.

Mohammad N. Akhter, director of the state's division of health, said one in five of his employes works to comply with federal regulation of the more than 40 health programs funded by Washington.

He predicted that his office could do a better job with 75 percent of the funds it now gets from Washington if the federal mandates were eliminated. "To have seed money, to have a push from Washington, that's great," he said. "But once a program is institutionalized, it's not cost-effective to run it from Washington. They don't know the difference between St. Louis and Boot Hill."

Eleven Missouri counties now have no public health units, and Akher and Toan would rather see such units established than money spent to assure that all parts of the state are served by doctors. "You can provide more basic services through a public health unit," Toan said.

Reagan's proposal to lump a number of health programs into one large block grant to the states presumably will provide the flexibility state officials want.

There are other examples of federal intrusion into state activities here, from environmental control. Bond and his fellow governors say they can solve these problems better and more cheaply without so much advice from Washington. Even if they can't, Bond said, the decisions should be made closer to home.

"We may make some mistakes in Missouri," Bond said, "but I kind of like the republican form of government, where you can fix responsibilities and people can change things if they don't like it. I don't see that there's any more wisdom coming out of Congress or the bureaucracy than out of the state capitals."