WASHINGTON — Senate Democrats plan to press ahead this week with an effort to push new voting rights protections through Congress, in an all but doomed attempt to enact a key piece of President Biden’s agenda that has been undercut by members of his own party.

The Senate on Tuesday will begin to debate legislation that combines two separate bills already passed by the House — the Freedom to Vote Act and the John Lewis Voting Rights Advancement Act — and folds them into an unrelated measure. The move would allow the Senate to bring the bill directly to the floor, avoiding an initial filibuster.

But that strategy would still allow Republicans to block it from coming to a final vote, and Democrats lack the unanimous support needed in their party to change Senate rules to muscle through the legislation themselves. Still, Senator Chuck Schumer of New York, the majority leader, said late last week that Democrats would forge ahead anyway, forcing Republicans to publicly declare their opposition to the bill.

“We all have to be recorded at this moment in time about where are we in protecting the right to vote,” Senator Tim Kaine, Democrat of Virginia, said on CBS’s “Face the Nation” on Sunday. “Right now, it doesn’t look like it has the votes to pass, but we’re going to cancel our Martin Luther King Day recess and be there this week because we think it’s so important for the country.”

The push to proceed even in the face of almost certain failure reflects the party’s conundrum, facing two key defections in its ranks and a wall of Republican opposition. It comes days after a critical Democrat, Senator Kyrsten Sinema of Arizona, emphatically announced that she would not support undermining the filibuster to pass legislation under any circumstances and Senator Joe Manchin III of West Virginia reiterated the same position.

Mr. Kaine suggested on Sunday that other paths around the filibuster existed, including narrowly altering it explicitly to pass the voting rights bill, and lengthening the debate time in an effort to pass the bill on a simple majority vote.

But privately, Democrats have been less sanguine, especially after a remarkable speech delivered by Ms. Sinema on the Senate floor on Thursday, just hours before Mr. Biden was scheduled to lobby Democrats on the bill. The speech, in which she declared unwavering opposition to altering the filibuster, sent a fresh wave of fury through the Democratic ranks.

“These two Democrats have decided that it is much more important to them to protect the voting rights of the minority on the Senate floor than to protect the voting rights of minorities in this great country of ours, the minorities that made it possible for them to be in the position that they’re currently in,” Representative James E. Clyburn, Democrat of South Carolina and the majority whip, said on ABC’s “This Week” on Sunday. “So, I hope, but I don’t think, that we will change their mind.”

A failed vote on the legislation threatens to become the Biden administrations second high-profile setback in about a month. In December, Mr. Manchin declared that he could not support the president’s sweeping social policy and climate bill as written.

But Mr. Manchin and Ms. Sinema’s opposition to weakening the filibuster to pass the voting rights bill has particularly irked other Democrats, who have cast the legislation as a remedy to an existential threat to democracy posed by voting restrictions enacted by Republicans across the country.

Speaking at a news conference in Washington on Monday with the family of the Rev. Dr. Martin Luther King Jr., Speaker Nancy Pelosi of California argued that “nothing less is at stake than our democracy” and issued a tart entreaty to Senate Democrats.

“If you really, truly want to honor Dr. King,” she said, “don’t dishonor him by using a congressional custom as an excuse for protecting our democracy.”

Martin Luther King III, Dr. King’s eldest son, invoked his father’s “Letter From Birmingham Jail,” in which Dr. King described the “white moderate” as Black Americans’ greatest “stumbling block in his stride toward freedom.”

“He was surrounded by people who told him to wait until a more convenient time and to use more agreeable methods,” Mr. King said. “Fifty-nine years later, it’s the same old song and dance from Senators Manchin and Sinema.”

The Freedom to Vote Act contains a slate of proposals to establish nationwide standards for ballot access, in an effort to counteract the wave of new restrictions in states. It would require states to allow a minimum of 15 consecutive days of early voting and that all voters are able to request to vote by mail; establish new automatic voter registration programs; and make Election Day a national holiday.

A second measure, named for Representative John Lewis, the civil rights icon who died in 2020, would restore parts of the landmark Voting Rights Act weakened by Supreme Court rulings. Among the provisions was one mandating that jurisdictions with a history of discrimination win prior approval — or “preclearance” — from the Justice Department or federal courts in Washington before changing their voting rules.

Republicans have uniformly opposed the legislation, casting it as inappropriate federal intervention in state voting operations and a partisan exercise intended to give Democrats an unfair advantage.