The government's program of free legal aid to the poor, which survived last summer's budget cutting, is now caught up in a complicated new dispute that could result in budget cuts through the back door.

At issue is a threatened effort by the new directors of the Legal Services Corp. named by President Reagan to review all the funding commitments for 1982 already made by the outgoing board members, who are holdovers from the Carter administration.

That effort could blow up into a major confrontation later this month, when the board must decide whether to send out about $20 million in March funding to legal aid groups across the country.

William Olson, the newly appointed acting chairman of the board, has said the new board will review the funding commitments made by Carter administration appointees. Reagan wanted the corporation abolished altogether, but after much battling in Congress last year, it survived with a $241 million budget for 1982, a cut of 25 percent.

LSC supporters were able to keep the program alive at least in part by reminding Reagan that he would have his own choice of board members and by agreeing to new restrictions on the corporation's activities designed to prevent it from engaging in politics, from lobbying and from filing broad-based, class-action lawsuits.

If any action is taken to cut the funding further, legal aid advocates are considering filing lawsuits charging that the new board members were appointed illegally since they were sworn in while Congress was in recess and have not yet been approved by the Senate.

The administration named the new members on the afternoon of Dec. 31 to replace seven of the 11 holdovers. Olson convened a board meeting so rapidly that six of the members had to participate by telephone. There was no public announcement and no member of the staff was present.

A resolution was adopted to prevent corporation President Dan J. Bradley from awarding any grants for 1982 until the new board could conduct a review of the award criteria.

What Olson did not realize at the time was that virtually all of the contracts for the 1982 calendar year had been approved at least a month earlier and that funding for January and February had already been sent out to the 350 local and state groups that provide legal services for poor people.

The checks for March are to be sent Feb. 1 unless the board takes some action to the contrary.

LSC proponents are especially unhappy that Olson, a 32-year-old Virginia lawyer with strong ties to the conservative wing of the Republican Party, may not have to go through a Senate confirmation hearing.

Olson headed the Reagan transition team assigned to the LSC, and there are reports that he recommended abolishing the corporation. Reagan himself has also long been an adversary of the legal services program.

Olson said in an interview that he has promised not to disclose the contents of the transition report and cannot comment. "This puts me at a disadvantage because people can speculate on what was in it," he said.

But Susan Kellock, executive director of the Equal Justice Foundation, has asked the administration to make the report public. "If it is true that Mr. Olson supported President Reagan's position to abolish the LSC, then the public has the right to know the contents of the team report and the position taken by its chairman," she said.

Administration sources say the final four board members are also likely to be appointed on a recess basis before Congress returns Jan. 26.

Steven L. Engelberg, one of the four remaining members of the old board, said, "I believe there are serious questions as to the legal propriety of making recess appointments. No one is challenging his authority to replace this board . . . . But he could have replaced the board in an orderly, legitimate fashion following the statutes. Instead, he waited for the congressional recess."

In fact, the terms of six of the board members had expired by the time Reagan took office and the remaining five expired last July. Administration sources have said Reagan did not move more swiftly to replace the Carter appointees because he hoped to eliminate the program completely.

Engelberg added, "The confirmation process is a shield against putting people on the board who are avowed enemies of the program . . . . This is not the Department of Energy. This is a program created by Congress. Reagan has a responsibility to appoint people pledged to uphold the law."

The Coalition for Legal Services, a private group interested in preserving free legal services for the poor, is considering filing a lawsuit. It has sent a letter to Olson saying that the appointments were not legally authorized and that the board therefore cannot make legally binding decisions.

The letter asked Olson "to refrain from acting as a director unless or until you have been appointed, confirmed and qualified as required by the Legal Services Corp. act . . . . "

Administration sources say some of the names may be sent to the Senate for confirmation, but the White House legal position is that the board members appointed during the recess have the authority to make decisions and can serve until the end of the current congressional session--the end of 1982--without Senate confirmation.

In its letter to Olson, the Coalition for Legal Services argues that although the Constitution allows recess appointments for "officers of the United States," the act creating the LSC explicitly says the corporation is not an "agency or department of the government and its officers and employes are not officers of the United States."

Olson conceded there is a "legal issue as to what extent the new board can affect the 1982 grants. We are trying to act slowly and deliberately and not get the cart before the horse. We're trying not to take actions that are premature."

Olson said he is operating on the assumption that the board has the authority to make substantive decisions, but he would not say whether he plans actually to do anything. "It's a good time to come in," he said. "You don't have to worry about making critical decisions in these months."