The mayors and the governors visited the White House yesterday, delivering to President Carter sharply divided messages on what kinds of cuts in the fiscal 1980 budget they could tolerate.

A delegation from the National Conference of Democratic Mayors, according to several of its members, told the president that the kind of deep cuts in social welfare programs he is contemplating could cause a "disintegration" of urban policy leading to "disatrous" consequences for the nation's big cities and possibly the Democratic Party.

The mayors were followed by a few minutes by a delegation from the National Governor's Association with a different message. The governors, according to Chairman Julian M. Carroll of Kentucky, told Carter they would accept the budget cuts, but only if the federal government consolidates many of its aid programs, giving the states more control, and delays implementing certain other costly programs.

The division between state houses and city halls was evident by the remark of Mayor Kevin White of Boston, chairman of the Democratic mayors group, who called the governors' proposal "just another way to get control of the funds at the [state] level.

"I'd fight that almost as much as i'd fight the cuts" in funds, White said.

Neither group received much of an answer from the president. White said Carter "listened" to the mayors' appeal and carroll said Carter asked the governors for specific examples of programs that could be consolidated or delayed but made no commitments.

The White House visits were part of the final scramble by various interest groups as the president makes his decisions on the 1980 budget, which he has promised will be "very tight" and which is expected to include substantial cuts in domestic programs. The coming congressional battle over the budget is already showing signs of straining traditional Democratic alliances.

Mayor Lee Alexander of Syracuse, N.Y., said the mayors will meet in Washington Jan. 23 and 24 and decide then whether to wage an all-out fight in the Congress against proposed administration budget cuts.

The White House is already gearing up for that expected fight. It plans to recruit up to six lobbyists from other government agencies to do nothing next year except monitor the congressional budget process and defend the president's proposals.

The mayors indicated yesterday that they will not give in to Carter's budget austerity plans without a battle. Mayor Kenneth Gibson of Newark said he told the president he has had to lay off 440 city employes, including 200 police officers.

"When we can stand up and make statements in support of shaky shahs, recognize Red China and two days later have an exclusive Coca-Cola franchise there, and I can't get a commitment on 200 cops in Newmark, then I have a concern about the priorities in the White House," he said.

White, who said up $15 billion in urban aid funds is at stake in Carter's budget decisions, warned that the big cities' constituencies "are essential building blocks for the Democratic Party" and could be alienated if the administration slashed social welfare programs.

For their part, the governors essentially offered the White House a trade-off: support for budget austerity in return for the consolidation and greater control over federal programs they have been seeking.

Carroll said the governors' support is "absolutely" contingent on consolidation administrative costs to the states. He cited, as an example of programs mandated by Congress that might be delayed, requirements to remove architectural barriers to the handicapped in public buildings, which he said would cost more than $50 million in Kentucky alone.

Carroll said that while the governors did not particulary oppose the mayors' appeals to go easy on cutting urban aid programs, "somebody has got to suffer" in the anti-inflation effort and "the mayors might as well get ready to suffer as well."

They mayors however, argued that they have already cut their own programs and that they came to the White House, in White's words, "to fight for jobs, for money that belongs to the cities."

White described the mayors' session with Carter as "a tough meeting, there was tension . . . There was a lot more tension in this meeting than in other meetings I've been in with other presidents."