The Senate yesterday dealt an apparently fatal blow to the Clinch River breeder reactor by voting, 56 to 40, to reject a last-gasp financing scheme to resurrect the imperiled nuclear power project.

The vote represented a sharp, although not surprising, personal loss for Majority Leader Howard H. Baker Jr. (R-Tenn.), who always before had managed to keep the controversial Tennessee project afloat despite repeated attempts by the House to scuttle it.

"One of these days I think we will regret not having an entry in this sweepstakes," said a somber Baker after the vote, referring to the technology under which breeder reactors would produce more nuclear fuel than they use to generate electricity. "But the Senate has spoken; I will not try to prolong the issue at this time."

Sen. James A. McClure (R-Idaho), who led the floor fight for a $1.5 billion appropriation that would have been matched by $1 billion from private sources, said that backers of the project will explore new financing alternatives but added, "I'm not very hopeful of those alternatives.

Said Sen. Dale Bumpers (D-Ark.), who teamed with conservative Sen. Gordon J. Humphrey (R-N.H.) to defeat the plan, "I take it is dead as far as the eye can see, at least I hope so."

A spokesman for Energy Secretary Donald P. Hodel said that Hodel was "disappointed" and added that the department will begin "orderly termination of the project."

The government has spent $1.7 billion on the project, including design, engineering and excavation work. But Congress, responding to arguments that the project is obsolete, dangerous and wasteful, voted earlier this year to end it unless private industry paid more of the cost.

The joint-financing scheme was proposed, largely at Baker's behest, as an amendment to a supplemental appropriations bill. It was that amendment that was tabled, or rejected, yesterday.

In arguments over the amendment, Bumpers ridiculed the contention that private investors would be shouldering any risks or bearing any real burden. Instead, they would be guaranteed a 37 percent return on their investment, Bumpers said, adding, "I have never heard of anything as shameless."

But, although Bumpers handed Baker one of the majority leader's rare defeats in the Senate, he praised Baker after the vote as "magnanimous" in his loss. "It's not ever easy to be a good loser," Bumpers added.

Baker had faced dwindling margins in his effort to keep the project alive, winning last year by a margin of one vote after considerable arm-twisting on the floor. But this time, as it became clear he would lose, Baker released some senators who would have voted with him if their votes had been crucial, and the margin turned out far bigger than either side had predicted.

Of Washington area senators, only Sen. Charles McC. Mathias Jr. (R-Md.) voted with Baker.

The project, located on the Clinch River near Oak Ridge, Tenn., was estimated when first approved in 1970 to cost $800 million. Its latest cost estimate was $4.2 billion, including the $1.7 billion already spent.

Regarding the joint-financing plan, worked out earlier this year by government and industry representatives, the Congressional Budget Office concluded in a recent study that under the plan the breeder would cost taxpayers $250 million more than if it were financed strictly by government funds.

President Reagan had backed the project and the cost-sharing plan to finance it, warning in a letter circulated to senators this week that it "truly would be ironic if, on this 10th anniversary of the oil embargo, during a time of heightened tension in the Middle East, we refused to complete this project at a cost equivalent to approximately eight days of imported oil."

But opponents argued that the project has become outmoded since 1970 and that it would lead to dangerous nuclear proliferation. They also said that its cost was excessive in relation to its benefits to consumers, with industry reaping big benefits at low cost.