The leadership of the nation's governors urged President Reagan yesterday to avoid further deep cuts in federal domestic spending and suggested that he look to the defense budget, entitlements and taxes for future savings.

"While we fully support your goal of strengthening our military defense capability," leaders of the National Governors Association said in a letter to Reagan they delivered to the White House, "we also believe that a sound economy, an adequate transportation system and a well-educated and highly trained work force are also essential to our national security."

Democratic governors met with the Senate Democratic leadership. A bipartisan group concluded the day with a meeting at the White House with top aides, stepping up the campaign against further cuts in federal grants-in-aid to state and local governments.

The rhetoric of opposition to more reductions was strong this week, with governors association president Richard Snelling (R-Vt.) calling the administration's policies an "economic Bay of Pigs." Concerned about this, Reagan personally telephoned two Democratic and two Republican governors Thursday. When he spoke with Illinois Gov. James R. Thompson (R), Reagan reportedly said lightly, "I'm just calling to see if anybody out there still loves me."

The governors made formal their request that Reagan convene a "domestic summit" of governors, mayors and congressional leaders with hard debate before the president decides on cuts in the fiscal 1983 budget.

The summit idea was greeted warily by presidential counselor Edwin Meese III when the leaders of the governors association met yesterday with him, White House chief of staff James A. Baker III, Office of Management and Budget Director David A. Stockman and Richard Williamson, presidential liaison to state and local governments.

Meese reportedly told them that he was concerned that such a meeting could easily turn into a "media event," raising expectations that could not be delivered.

Stockman reportedly told them that it would be difficult to promise that they would be spared further cuts. He said the drop in the inflation rate combined with the recession had produced an unanticipated revenue shortfall of $30 billion.

Before the White House meeting, several of the 15 visiting governors spoke with reporters about their growing problems.. They expressed the fear that they would again be called upon to bear the brunt of cutbacks the president is expected to announce next month.

"I think the governors are furious. There's a great undercurrent of feeling of having been had," said Gov. Jay Rockefeller (D-W.Va.). "We're being asked to make all of the sacrifices and we're sick of it. What they're doing is gutting the tax structure of this country in behalf of business and asking the states to make up the difference."

Republican governors, including Snelling, tended to sound the same theme, if more muted. "If Congress has to reduce growth in spending, it ought to spread the misery," said Tennessee Gov. Lamar Alexander, a Republican.

At breakfast in the Capitol, 11 Democratic governors met with the Senate Democratic leadership to discuss the proposed cuts and legislation to shift responsibility for federal programs to the states. These are bills that Democrats have strongly opposed.

This was the first such meeting on federalism; participants considered it significant not so much for what it accomplished but because it occurred at all. Asked why they had never met before, Kentucky Gov. John Y. Brown said, "I think we need each other more now."

Added Senate Democratic leader Robert C. Byrd (W.Va.): "There wasn't perceived to be the need at the time we were in the majority. We're in the minority now. We're learning to live in the minority."

The senators promised the governors that they would meet again in February and would support legislation to require that bills in Congress include "fiscal notes" calculating the added costs to state and local governments.

There was no agreement on the question of ceding new power to the states, however. When that matter came up, Utah Gov. Scott M. Matheson recalled, the senators were "conspicuously silent."