WASHINGTON — Democrats pushed forward on Tuesday with what appeared to be a futile bid to enact voting rights protections over Republican opposition, taking up legislation that they said was urgently needed to counter widespread balloting suppression efforts and laying out a plan to try to steer it around G.O.P. obstruction.

Using a procedural shortcut, Senator Chuck Schumer, Democrat of New York and the majority leader, avoided a Republican blockade that has stalled the legislation for months to force it to the floor. But Democrats were far short of the votes needed to win its passage over Republican opposition, and lacked the votes needed in their own party to change Senate rules and enact it unilaterally.

Still, they announced that they would mount a long-shot effort to establish an exception to the filibuster for voting rights bills, requiring opponents to hold the floor for an old-style “talking filibuster” that would allow a final, 51-senator majority vote — instead of the 60 now needed — to move forward after all senators had exhausted their opportunities to speak.

“If the Senate cannot protect the right to vote, which is the cornerstone of our democracy, then the Senate rules must be reformed,” Mr. Schumer said.

The Democrats’ plan, unveiled in a private party meeting on Tuesday night, would still require a partisan vote to change the rules, meaning it cannot succeed at this point given resistance from at least two Democrats.

No Republican currently supports the voting rights measure, which combines two far-reaching bills intended to protect access to the ballot box, leaving Democrats 10 votes short in the evenly divided Senate.

The impasse has led to intensifying calls to unilaterally change filibuster rules so Democrats can bulldoze over Republicans’ objections. But at least two Democrats, Senators Kyrsten Sinema of Arizona and Joe Manchin III of West Virginia, have made clear they will not do so, even though they support the bills.

Their positions, reinforced by Ms. Sinema in a floor speech last Thursday, have drawn sharp criticism from activist groups who say their stance will give Republicans a free hand to rewrite voting rules in states around the country.

“What good is preserving a dysfunctional tradition of bipartisanship if bipartisanship cannot even preserve democracy?” the N.A.A.C.P. wrote in a letter sent to Democratic senators on Tuesday. “It is morally inconsistent to praise voting rights legislation while allowing a procedural rule to tank it.”

The reluctance of the two senators has enraged Democratic activists and the party’s progressive base, leading Emily’s List, a prominent political action committee that backs women who support abortion rights, to threaten on Tuesday to drop its support for Ms. Sinema if she maintained her position. Senator Bernie Sanders, independent of Vermont, said he would consider backing primary challengers to the two holdouts.

Though the effort seemed preordained to fail, Mr. Schumer has made it clear that he will still pursue a vote and a rules change. Should that push fall short as expected, Democrats want all senators to be on the record on the issue, and to demonstrate to their progressive allies, campaign donors and Democratic voters that they did all they could to try to protect voting rights.

“Win, lose or draw, members of this chamber were elected to debate and vote, especially on an issue as vital to the beating heart of our democracy as voting rights,” Mr. Schumer said. “The public is entitled to know where each senator stands on an issue as sacrosanct as defending our democracy.”

The Democratic push has drawn a scathing response from Republicans, who portray it as a power play to gain more control over state-run elections while undermining a procedural tool they say is fundamental to the nature of the Senate.

“A faction this desperate for unlimited short-term power is a faction that must be denied it,” Senator Mitch McConnell, Republican of Kentucky and the minority leader, said on Tuesday as he accused Democrats of hypocrisy for backing the filibuster when they were in the minority but seeking to overturn it now.

The legislation, which could face a vote as early as Wednesday, consolidates two measures that had previously been approved by the House but blocked from consideration in the Senate — the Freedom to Vote Act and the John Lewis Voting Rights Advancement Act. If they were approved by the Senate in their present form, they could be immediately signed into law by President Biden.

Under Democrats’ current strategy, once they try and fail to break the filibuster, Mr. Schumer would move as early as Wednesday evening to change the rules. Success in changing the rules would require the support of all 50 Democrats and independents, plus the tiebreaking vote of Vice President Kamala Harris.

But Mr. Manchin, who has voiced support for the talking filibuster, said again on Tuesday that he would not back a partisan effort to change the rules.

“I just don’t know how you break a rule to make a rule,” Mr. Manchin told reporters before the meeting.

The Freedom to Vote Act includes an array of proposals to establish nationwide standards for ballot access, aiming to nullify the wave of new restrictions in states. It would require a minimum of 15 consecutive days of early voting and that all voters be able to request to vote by mail. The measure would also establish new automatic voter registration programs and make Election Day a national holiday. It is a narrower version of legislation that Democrats introduced early last year but was revised to suit Mr. Manchin, who said the original bill was overly broad and insisted on including a provision requiring voters to present some form of identification.

The 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.

As the debate began, Democrats and Republicans took the floor to try to make their respective cases even with the likely defeat of the measure. Republicans reiterated their deep offense at Mr. Biden’s speech in Atlanta last week in which he compared opponents of the bill to notorious Southern racists such as Bull Connor and George Wallace of Alabama.

Senator Dan Sullivan, Republican of Alaska, noted that New York voters last year rejected proposals to ease voting in ballot initiatives.

“If the majority leader keeps coming down calling Republican states that restrict voting Jim Crow 2.0, is he going to go to Times Square and call his own constituents Jim Crow 2.0?” Mr. Sullivan asked.

Democrats said Senate Republicans were looking the other way as Southern states sought to crack down on voting access after Democrats won the presidency and the Senate majority under 2020 voting rules that were instituted in some states because of the pandemic.

“We Democrats cannot sit back and let 2020 be the last free and fair election in our country,” said Senator Mazie K. Hirono, Democrat of Hawaii. “If we don’t protect the right to vote, we won’t have a democracy. It’s that simple — that’s the reality.”

Some activists urged the Senate not to end the debate too quickly and to engage Republicans in an extended fight, noting that backers of the 1965 Voting Rights Act forced opponents to maintain a filibuster for nearly 60 days before it was broken.

“The Senate must not spend a couple of days on this battle and then move on,” said Fred Wertheimer, the president of Democracy 21. He said Democrats “must force the supporters of the state voter suppression and election sabotage laws to engage in a talking filibuster and try to justify their position.”