Congress, even while putting the finishing touches on nearly $40 billion in spending cuts, showed signs of flinching yesterday on one of them: a decision by both houses to eliminate the $122 per month minimum Social Security benefit.

Leading the charge to restore the benefit now received by about 3 million retirees, House Majority Leader James C. Wright (D-Tex.) proposed late Thursday that the House instruct its conferees on the spending cut bill to make sure that the benefit is retained for those now receiving it.

Yesterday, fearful of derailing the smoothly running conference, House Minority Leader Robert H. Hichel (R-Ill.) responded with a charge of "sabotage," and Budget Committee Democrats said the issue should be handled separately from the conference.

But, by the time the dust had settled, it was apparent that many Republican as well as Democratic lawmakers were having second thoughts about elimination of the benefit, from which President Reagan hopes to save $1 billion next year.

"They're really beginning to feel the pinch. It's an issue out there and it's growing . . . people look at this and think Reagan wants to take away their Social Security too," said a Republican leadership aide.

The minimum benefit establishes a floor assuring recipients at least $122 a month even if their earnings records would otherwise entitle them to less. The White House contends that it is essentially an "unearned beenfit," while its defenders describe it as the margin of survival for thousands of the elderly poor.

The Senate Budget Committee has estimated that 1.2 million of the 3 million recipients would be eligible for other federal benefits if the minimum is eliminated, leaving 1.8 million to suffer varying degrees of income loss. The Office of Management and Budget figures the average monthly loss at $62.

The battle is over repeal of the benefit for current recipients. Even House Democrats proposed an end to the benefit for future recipients. The versions passed by both houses would eliminate it for everyone, the Senate immediately, the House as of next April.

Although a White House official said the administration remains committed to elimination of the benefit, Senate Majority Leader Howard H. Baker Jr. (R-Tenn.) said "perhaps" the benefit should be restored in legislation considered after passage of the spending cuts package.

A Senate Republican source said there has been "preliminary talk" of restoring the benefit as part of a broader initiative on Social Security, adding that as many as a dozen Republican senators have approached the leadership on the matter.

Republicans on the House side also reported some anxiety about repeal of the minimum benefit, saying there has been talk of restoring it, through a means test, at least for the poorest of current recipients. An aide to Michel said the GOP leader "would not be inclined toward full restoration, but he's not opposed to dealing with it in a separate bill after the conference."

Wright's proposal remains before the House in the form of a resolution urging that "the necessary steps be taken to insure that Social Security benefits are not reduced for those currently receiving them."

It is expected to come up for a vote Tuesday, when, by coincidence or otherwise, the National Council of Senior Citizens is planning a massive rally on the Capitol steps to protest Reagan's proposed cutbacks in Social Security.

In comments yesterday, Wright continued to interpret the resolution as meaning the conference should retain the minimum benefit, despite a joint leadership "understanding" that the conference would not change provisions on which both houses agreed.

Although Michel called Wright's proposal a "breach of faith" that violated the understandings aimed at producing a successful conference, Baker seemed less disturbed when he appeared with House Democrats to discuss what was described as good progress in the conference. While the issue shouldn't be reopened in the conference, Wright was just "doing what we all do from time to time" in capitalizing on a good political issue, said Baker.

Even though Wright's resolution carries no force of law, it would achieve a longstanding Democratic goal of forcing a separate vote on controversial Reagan spending cuts. Wright did not rule out the possibility of similar moves on other issues.