Senate backers of a major trade bill are expecting an uphill battle to find enough votes to override President Reagan's veto, in striking contrast to the quick and overwhelming House vote to reverse the president's action.

"We have to have another three votes at this point and I don't know where they are going to come from," said Senate Finance Committee Chairman Lloyd Bentsen, D-Texas, a chief sponsor of the 1,000-page trade measure.The Senate is expected to take up the legislation after it returns from the Memorial Day recess.

Reagan's veto of the legislation on Tuesday came as no surprise. He has been critical in numerous speeches of a provision that would require companies to give employees 60-days notice of plant closings and large-scale layoffs. He has said that businesses need flexibility to survive hard times.

In his veto message to Congress, Reagan repeated that argument.

"I support voluntarily giving workers and communities as much advance warning as possible when a layoff or closing becomes necessary. . . . It is the humane thing to do.

"But I object to the idea that the federal government would arbitrarily mandate, for all conditions and under all circumstances, exactly when and in what form that notification should take place."

The president urged lawmakers to submit a revised version trade bill that he could sign in good conscience.

Within hours of Reagan's veto, the House voted 308-113 to override it. That was far more than needed to be successful.

Sixty Republicans broke with Reagan to vote for the override, while only one Democrat, Robert J. Mrazek of New York, supported the president.

The plant-closings measure is among scores of provisions contained in the bill. The main ones would overhaul the U.S. system of import restrictions. Others range from repealing the windfall profits tax to adding billions of dollars in agricultural subsidies.

Democratic leaders have argued that patching together a stripped down trade bill might be tough before Congress leaves town for the campaign trail.

Uncertain of override prospects in the Senate, many Democrats focused on making the debate a campaign issue.

"I happen to think it's a better issue for Democrats than for Republicans," House Speaker Jim Wright, D-Texas, said.

Michael Dukakis, the front-runner in the battle for the Democratic presidential nomination, called the president's action "unconscionable."

He said that Reagan and Vice President George Bush "don't understand that for working families who struggle to pay the mortgage, save for their children's education and care for their elders, there is nothing more painful than losing a job."

He contrasted the veto to a tentative deal under which drug charges against Antonio Manuel Noriega, the Panamanian leader, would be dropped if he agreed to relinquish power and leave Panama by August.

"Any administration that is willing to give General Noriega 120 days notice and a plea bargain ought to be willing to give the American worker 60 days notice before they are thrown out on the street," Dukakis said.

Reagan said that he vetoed the measure "with sincere regret," but said in his message that returning the measure to Congress was necessary.

"I am convinced this bill will cost jobs and damage our economic growth," the president said.

He said that the American economy has created 16 million jobs during his administration and that the unemployment rate is at its lowest point in 14 years.

He also said that the measure would "push us in the direction of protectionism. Closing our borders is not the solution to opening foreign markets," Reagan said.

Administration supporters rallied to Reagan's support following the veto.

Rep. William Frenzel, R-Minn., scoffed at the image of "unfeeling and mean-spirited owners of large and small American businesses putting upon weak and defenseless workers who are about to be thrown out on the street . . . unless big government puts its firm and fatherly hand" on the situation.

"As usual, this House believes that big government knows more than we in private life," Frenzel said. "This debate paints a depressing view of America and it is inaccurate."