President Reagan cautioned the nation's governors yesterday against pressing for costly federal solutions to welfare and child-care problems.

His remarks to the National Governors' Association were labeled a disappointment by Massachusetts Gov. Michael S. Dukakis, a Democratic presidential candidate, who said Reagan should be backing

the governors' bipartisan welfare-

revision bill in the Senate.

Reagan instead said he strongly supports a less costly Republican alternative rejected by the House two months ago.

The president also cautioned the governors to go slow on child care. Congressional Democrats and Republicans are proposing to expand federal support and subsidies for working parents.

"Maybe it's my conservative bent, but I can't help but feel uneasy sometimes," he told the governors gathered at the White House.

Welfare programs have become "a crippling poverty trap . . . . They keep the poor poor," Reagan said. "Now, much of the push for child care is designed to rectify the ills of earlier programs, and many of these efforts are timely and good. But in this area, more than any other, government should tread carefully."

He said the alternative bill sponsored by House Minority Leader Robert H. Michel (R-Ill.) would also give states freedom to "develop your own ideas on child care."

Afterwards, Dukakis told reporters that he was very disappointed.

"Just one word from the president, I think, would get this bill passed and done before the election, and get it off the table as a partisan issue and get going," Dukakis said.

"He keeps telling us stories about California, and we're trying to explain to him that the vast majority of people on welfare these days are single mothers with young children, and it's a combination of real training for real jobs, combined with day care, that makes a difference," he said.

White House chief of staff Howard H. Baker Jr. said Reagan would veto the House-passed welfare bill but might approve a less expensive measure being shaped by the Senate Finance Committee.

The governors are pushing a bill by Sen. Daniel Patrick Moynihan (D-N.Y.). It would toughen enforcement of child support, help states pay for job training and make participation mandatory for most unemployed adults. It would also provide some transitional Medicaid and welfare benefits for families exiting the relief rolls for work.