Senate Majority Leader Robert C. Byrd (D-W.Va.) yesterday called on President-elect Ronald Reagan to help salvage the fair housing bill. Byrd blamed "far right" Republicans for thwarting what he called a bipartisan coalition in favor of the legislation.

But it was a heavily lobbied group of conservative Democratic senators, mostly from southern and border states, who appeared to hold the key to the bill's fate late in the day as supporters attempted to tie down the 60 votes needed to keep the legislation alive in a crucial test today.

A half dozen or more votes apparently hinged on negotiations between the Democratic conservatives and Senate Judiciary Chairman Edward M. Kennedy (D-Mass.), chief advocate of the bill, that continued into the evening and then broke off until this morning.

"I'm still hopeful. . . . A lot of us really want a bill," said Sen. David L. Boren (D-Okla.) as the Kennedy forces drafted a new language for the conservatives to consider today.

After a week-long filibuster that forced Congress to drop its adjournment plans last Friday, the Senate finally agreed to a showdow cloture vote today on the bill, which would give the government broad new powers to enforce the 1968 legislation banning discrimination in housing.

The conservative Democrats helped provide 62 votes in favor of cloture in a preliminary vote last week but some have been wavering and were actively courted yesterday with offers of concession by both sides.

Byrd said Friday that he will withdraw the bill if cloture fails today. But if cloture is approved, he said, he will keep the Senate in session for as long as it takes to get final action.

"If we can keep them, or most of them, along with some absentees [from last Thursday's preliminary cloture vote], then we've got it," said one Democratic supporter of cloture.

However, another source said nose counts indicated that cloture might fail largely because many senators, including some who supported it before, feared a long list of post-cloture amendments could keep them in Washington until Christmas.

Sen. Orrin G. Hatch (R-Utah), leader of the filibuster forces, conceded late yesterday that "the odds are in their [the bill's advocates] favor."

Even if Byrd wins cloture today, Hatch said he and his allies have "well over 200 amendments" that could take the Senate a week or more to debate. "We wouldn't finish this session until Christmas," he said.

The legislation would empower the Department of Housing and Urban Development to bring suit to end discrimination in housing transactions, with fines of up to $5,000 for violators. Sponsors have already agreed to modify House-passed legislation to require that cases be argued before federal magistrates rather than administrative law judges. A further compromise under consideration last night, offered by Byrd and Kennedy, would also provide jury trials for damages, although not for injunctive relief, and would stake out an apparent middle ground in the controversy over whether intent or effect should be the test for determining guilt.

Byrd's appeal to Reagan followed similar urgings during the campaign from the Carter-Mondale camp for Reagan to take a stand on the issue. Although Reagan did not do so publicly, former senator Edward W. Brooke (R-Mass.), the only black member of the Senate in the last Congress, and now the chairman of the National Low Income Housing Coalition, said Reagan told him during the campaign that he favored fair housing legislation with some reservations.

Byrd cited Reagan's post-election endorsement of tax cut legislation as a precedent for lending support to the anti-filibuster drive on the housing bill. pBut Brooke said he doubted Reagan would get involved in the dispute.

Byrd's bid to Reagan, coupled with the apparent failure of the Democrats' efforts to work out a compromise with Hatch and the other conservative Republican filibusterers, put a new political gloss on the dispute. If Democrats lose the cloture vote or lose the race with the calendar on post-cloture amendments, they can blame the incoming Reagan administration as well as conservative Republicans for the loss of what they have described as the major civil rights legislation of a decade.