A bipartisan group of governors yesterday agreed to seek their colleagues' endorsement of a plan to trim the federal budget deficit, despite a heated meeting with President Reagan in which he denounced their call for a slowdown in defense spending and objected to criticisms of the "fairness" of his domestic cuts.

Utah Gov. Scott M. Matheson (D), chairman of the National Governors Association, said that when 45 members of the group met with the president yesterday morning Reagan "came out swinging" against their recommendation for about $294 billion in defense cutbacks or higher taxes in the next five years.

This is the key, and most controversial, part of the governors' budget blueprint, which is supposed to cut the projected federal deficit from $267 billion to $90 billion by 1988. The governors are scheduled to vote on it today.

Illinois Gov. James R. Thompson (R), vice chairman of the association, said that Reagan "took strong issue" with the proposed defense slowdown, but added that he told the president "it is not an attack on you or your policies."

Thompson said that he would continue to seek support for the resolution among the other 15 Republican governors, while Matheson "fiddled" with the language to try to satisfy Democrats, who occupy 34 statehouses, in order to persuade them that it is not too watered down to be useful.

One Democratic critic, Wisconsin Gov. Anthony S. Earl, circulated a substitute resolution that was even tougher on defense and more critical of Reagan's domestic priorities. The resolution, which Earl said has the support of at least five other governors, also calls for canceling the administration's July l tax cut.

Thompson said that if any such language was added "it's not going to pass" at today's plenary session, where a two-thirds majority is needed.

The unusually spirited debate among the governors, and between them and the president, was the result of the Matheson-Thompson effort to get the association involved in the broad issues of federal budget policy. In the past, the governors have lobbied mainly on issues in which the states are directly involved, particularly the federal grant-in-aid programs.

Efforts to find a compromise centered on some mild amendments, drafted by South Carolina Gov. Richard W. Riley (D), that expressed concern for "the millions of Americans who need work" and declared that any tax hikes should "not unfairly burden lower- or middle-income Americans."

Behind all the verbal maneuvering was the basic political question of whether the governors can maintain a bipartisan front when many of the numerically dominant Democrats want to make it clear that the group is repudiating Reagan's budget priorities.

Thompson insisted that it is not such a repudiation, and he was supported yesterday by Sen. Bob Dole (R-Kan.), who told the National Association of Counties: "I agree with them the governors on defense spending. I think we can budget a little better with cuts in that area."

But Democratic governors continued yesterday to voice the kind of criticisms of Reagan's budget that Thompson said had brought strong complaints from the president at their meeting.

"He's up to his keister with those criticisms," Thompson said.

But New York Gov. Mario Cuomo (D) insisted upon underlining the message.

"The governors think the president is wrong," he said. "Too much money is going into defense. The deficit is too large. We're saying: 'Let's be nice about it, but you Reagan are wrong.' "

With many states in deep financial crisis, several Republicans expressed the same frustration with the Reagan administration as did their Democratic colleagues. When Treasury Secretary Donald T. Regan met with one governors' committee at an afternoon session, the loudest words of criticism came from South Dakota Gov. William J. Janklow, a conservative Republican.

Janklow engaged Regan, who had given the governors a rosy picture of the nation's economy, in a bitter shouting match in which the South Dakotan accused the Reagan administration of shifting tax burdens to the states and looking out only for the concerns of corporations. When Regan said that the prime interest rate has dropped to 10 percent, Janklow retorted that this rate is not available to most consumers.

"We're looking for low interest rates for people, not for the corporations," he said.

In an impromptu debate with Earl and North Carolina Gov. James B. Hunt Jr. (D) Reagan referred, as he often has before, to the number of unfilled jobs in the want ads. Michigan Gov. James Blanchard (D) said, "It's the same story I've heard before. It's designed to convince us everything is fine."

Texas Gov. Mark White (D) called the help-wanted argument a "fake and deceptive approach to the problem of unemployment." White said that even though Texas is a major defense contractor, he supports the slowdown in defense spending that is urged in the draft resolution.

Other, less controversial, portions of the resolution endorse the Social Security compromise, call for a slowdown in most entitlement spending, "almost full funding" of need-tested programs, and support pegging the growth of federal aid to states at 75 percent of the national rate of inflation.