Senate Majority Leader Robert C. Byrd (D-W.Va.) said yesterday that if President-elect Ronald Reagan nominates Alexander M. Haig Jr. to be secretary of state the Senate will take a "very close look" at Haig's role in the Watergate affair.

"I'm not going to presume what the president-elect should do, that's his responsibility," Byrd said at his weekly news conference in his Senate office. "But the Senate has its responsibility and if his name [Haig's] is submitted, the Senate will do its duty, even if it means denying the president his choice."

Byrd's remarks further dampened the enthusiasm of the Reagan camp for Haig at a time when that enthusiasm was already waning. Sources in the Reagan camp said yesterday that Haig's nomination as secretary of state was increasingly doubtful. Said one source: "The problems are substantive, we know that." Said another: "The trend is certainly to look for someone else."

The Reagan team may already have approached former treasury secretary George P. Shultz and former health, education and welfare secretary Caspar W. Weinberger about taking the secretary of state job. Shultz reportedly has told Reagan he does not want a Cabinet appointment. Weinberger was one of the candidates for state before Haig's name surfaced in the last week as the leading candidate.

"I've always been willing to bend over backward a little bit on a new president's Cabinet choices," Byrd said. "However, it would be irresponsible of the Senate if Gen. Haig's name is submitted not to take a very close look at his Watergate role. I think the questions raised are serious enough that it is possible he would not be confirmed."

Byrd declined to specify what Watergate roles Haig played that raised these questions except to say that many of them were detailed in published press accounts. The press has said that Haig played roles in the pardon granted by then-president Ford to former president Nixon, in the wiretaps put on the telephones of 12 reporters and public officials by the Nixon administration in 1969-71, in the efforts by the Nixon White House to obstruct the Watergate investigation and in the withholding from the special prosecutor's office of the tapes of Nixon's Oval Office conversations.

In the published testimony, Haig explained the celebrated 18 1/2-minute gap in one Nixon tape as a "sinister force" that "had come in and applied [another] energy source and taken care of the information on that tape."

Haig served Nixon as White House chief of staff. When Ford succeeded Nixon after Nixon resigned in 1974, Haig became commander of the North Atlantic Treaty Organization in Brussels. Haig is now president of United Technologies Inc.

Byrd said that Haig's nomination would be questioned in the Senate even though the Republicans will be in the majority.

"I'm saying that as one senator who served in the Senate during that dismal episode [Watergate] that without taking any head count Gen. Haig's role will be examined," Byrd said. "There are no ifs, ands or buts about it."

On the upcoming cloture vote in the Senate over the fair housing bill, Byrd said he had met twice on Friday with three Republican senators he would not identify and several Democrats, including Sen. Edward M. Kennedy of Massachusetts, to plan their strategy for the vote.

"I also talked to Sen. [Howard] Baker [the minority leader] and found him positive and receptive," Byrd said. "We've decided we will take up the cloture motion immediately after a motion to proceed on Tuesday."

The cloture vote will be taken to limit further debate on the fair housing bill, which has already passed the House and which has been filibustered by a number of Republican senators Byrd described as members of the "far right." One senator opposing the bill is Sen. Orrin G. Hatch (R-Utah), who has said he would keep filibustering the bill to prevent its coming to a vote on the Senate floor.

"There will be back-to-back votes on Tuesday, if we win the cloture vote," Byrd said. "The tougher one is the cloture vote, we need at least 60 votes for cloture to pass."

Byrd said if cloture fails he will move to withdraw the fair housing bill from Senate consideration this term, which he conceded will dim the chances for passage any time in the years immediately ahead.

"If this legislation is not passed, it should be eminently clear that a truly fair and just fair housing bill will not be enacted for some years to come," Byrd said. "The numbers of far-right Republicans will soon greatly increase and the outlook for progressive legislation is not something you can let out a hoot-and-a-holler about."