Backers of a constitutional amendment that would give District residents the right to select their senators and representatives began their final push yesterday to win passage of the bill, visiting uncommited senators and calling on President Carter and Vice President Mondale for help.

Both supporters and opponents of the bill predicted yesterday that the bill's fate would be decided by a handful of still undecided senators in Tuesday's scheduled vote.

As Senate and White House lobbyists for the bill were zeroing in on five undecided senators whose votes they considered especially critical, D.C. City Council Chairman Sterling Tucker and the city's Republican Party Chairman Paul Hays stalked Senate hallways in search of other uncommitted Senators.

"When are you going to allow right turns on red lights?" asked a legislative assistant to Sen. Dale Bumpers (D-Ark.) during the Tucker group's visit. "The senator gets angry everytime he must stop for one."

In the office of Sen. William Proxmire (D-Wis.), Tucker was told by one Proxmire aide that "he is going to vote with you, but if you overlobby him you might lose his vote."

The Tucker group quickly thanked the aide and left.

Tuesday's vote is the closest te District - in almost a century of trying - has come to winning an elected voice in Congress. But even if the bill is approved by the Senate (it has already been adopted by the House) it must still win the approval of at least 38 state legislatures over the next seven years before it becomes effective.

Some of the bill's backers and opponents say that could be a long and potentially unsuccessful battle.

The names of the senators whose votes are considered uncommitted was a closely held secret yesterday.

But one senator who asked not to be named identified these senators are targets of lobbyists for both sides:

Robert J. Dole (R-Kan.); Lloyd M. Bentsen Jr. (D-Tex.); Jennings Randolph (D-W. Va.); Herman E. Talmadge (D-Ga.); James R. Sasser (D-Tenn.); Thomas J. McIntyre (D-N.H.); Dennis DeConcini (D-Ariz.); John J. Sparkman (D-Ala.) Sam Nunn (D-Ga.) Mike Gravel (D-Alaska) and Bob Packwood (R-Ore.).

Bentsen, earlier listed as firmly opposed to passage of the bill, could not be reached for comment yesterday on the report that he is uncommitted.

A Washington Post survey yesterday indicated that, of those 11, at least Talmadge, Sasser and Gravel appear to be leaning toward approval of the bill. Nunn was to fly with President Carter to Georgia yesterday and was expected to be lobbied on behalf of the bill during the trip, one source said.

A White House source said lobbyists there would be concentrating efforts most heavily on Bentsen; Mark O. Hatfield (R-Ore.), reported earlier to be leaning toward approval; Nunn, and Dole, plus Pete V. Domenici (R-N.M.) and John G. Tower (R-Tex.), both of whom had been described as committed to opposing the bill.

Aides to Sen. Edward M. Kennedy (D-Mass.), who is leading the floor fight for passage of the bill, declined to discuss which senators they hope to sway. But one source said the Kennedy forces had 60 to 62 fairly firm votes for the bill. The most needed in any vote is 67.

Coupled with likely declarations of support from Talmadge, Sasser and Gravel, a successful lobbying effort by the White House could ensure victory for the bill's backers.

"We need between six and seven more votes, we figure," said one source who asked not to be identified. "But they think they've got what they need, and we think we've got the 67 (votes)."

"I don't think they have 60 votes," responded Sen. Orrin G. Hatch (R-Utah), one of the leaders in the fight against the bill. "It's a touch-and-go situation. I think we need two to five votes and we'll win it."

While Tucker and the others made their rounds of senatorial offices yesterday a block away in a small room in the Methodist Building five women busily telephoned black elected officials in 10 states.

These officials were being asked to wire or call their senators and urge them to support the amendment.

Mayor Walter E. Washington, who did not visit the Capitol yesterday, said he had been given a list of 12 senators and was asked to call them by the coalition leading tha fight locally for the bill. He said he would make the calls during the weekend.

About 35 people trooped through Senate halls yesterday, visiting about 30 offices to lobby for the bill.