WASHINGTON — President Biden’s endorsement on Thursday of substantially overhauling the filibuster was the clearest recognition yet that the Senate’s signature procedural weapon has rendered the polarized chamber a legislative wasteland, and that Democrats plan to try to do something about it.

But in lending his approval to an effort to “fundamentally alter” the filibuster, in the president’s words, he also acknowledged that Democrats must first wrap up a sprawling budget agreement that is carrying the bulk of his legislative agenda — a measure that itself can advance only because of special rules that shield it from a filibuster.

The president’s comments showed for the first time that he and other top Democrats view taking on the filibuster as too explosive at a moment when they are working feverishly to strike a budget deal with two Democratic senators, Joe Manchin III of West Virginia and Kyrsten Sinema of Arizona, who have been the principal objectors to changing the rules. Democrats fear that pushing ahead on both fronts simultaneously would alienate the two and potentially result in no progress on either front.

In the meantime, Mr. Biden said he would encourage Senate Democrats to force Republicans blocking their initiatives to appear on the floor of the chamber and defend their stance — and also to enforce a centuries-old but seldom applied rule that limits senators to speaking only twice a day on a given subject before they can be cut off. Under current practice, senators need not even show up to carry on a filibuster; it is up to the proponents of legislation to deliver the 60 votes needed to break the deadlock.

“It used to be you had to stand on the floor and exhaust everything you had,” Mr. Biden said during a town hall event on CNN.

Mr. Biden’s comments energized anti-filibuster forces stung this week when Republicans used the tactic for the third time this year to block voting rights legislation that Democrats say is crucial to preserving voting access, particularly for those in minority communities.

While Mr. Biden would technically have no role in changing Senate rules, his blessing was seen as essential given his 36 years in the chamber and his often voiced reluctance to disturb the traditions of an institution he reveres.

“To have a president with a 30-plus-year history in the Senate start making the case against the filibuster is truly historic,” said Adam Jentleson, a former top Democratic Senate aide and the author of “Kill Switch,” a book on the history and abuse of the filibuster. “Once you’re in, you’re in. As the president presiding over a democracy in crisis, you can’t say the filibuster needs to be scrapped and declare the assault on voting rights an existential threat, and then just sit back and let the chips fall where they may.”

While some lawmakers and activists have suggested a filibuster carve-out for constitutionally grounded issues such as voting rights or raising the federal debt ceiling, Mr. Biden raised the possibility of extending the reach of filibuster changes to other issues.

The filibuster has proved an increasingly insurmountable obstacle for a rash of proposals on issues such as immigration, gun control, police conduct and even the investigation of the Jan. 6 assault on the Capitol.

Once employed relatively rarely, it has become an everyday fixture of the Senate, requiring lawmakers to deliver a three-fifths supermajority — 60 votes — in order to move forward on virtually any major legislation. In a highly partisan 50-50 Senate, that has become a very tall order, threatening much of the Democratic agenda outside of measures that can be incorporated in the budget bill, which is protected from a filibuster under a set of complex rules that severely constrain what can be included.

Republicans say frequent use of the filibuster is justified because Democrats are pursuing a far-left agenda well beyond the reach of the narrow majorities they currently hold.

In recent days, Senator Mitch McConnell of Kentucky, the Republican leader and a leading practitioner of the filibuster, has handed Democrats a potent piece of evidence that they cannot accomplish anything of significance in today’s Senate with the filibuster in place.

Even as he has argued that it is imperative that Congress raise the federal debt ceiling to avoid a catastrophic government default, Mr. McConnell has insisted that Democrats use the filibuster-proof budget process to do so unilaterally, effectively conceding that as long as Republicans have the power to block the move, they will. His demand essentially amounts to an ultimatum for Democrats: Stop us before we filibuster again.

The debt ceiling fight was resolved with a temporary extension into December, but the showdown shifted the calculus of the filibuster fight. Democrats say that even their most reluctant members could be persuaded to ditch the filibuster if forced to choose between a worldwide economic calamity or preserving an arcane Senate rule.

Mr. McConnell seemed to recognize that possibility and backed down for the moment, but Democrats say the Republican recalcitrance shined a bright light on the need for change.

Mr. Biden on Thursday called the Republican blockade of the debt ceiling increase bizarre and said that if Republicans repeated the tactic, “I think you’ll see an awful lot of Democrats being ready to say, ‘Not me. I’m not doing that again. We’re going to end the filibuster.’”

To further build a case against the tactic, Senator Chuck Schumer of New York, the majority leader, plans next week to force a showdown on a separate voting rights measure — one named for the civil rights hero John Lewis. It would restore elements of the Voting Rights Act that have been undermined by unfavorable Supreme Court rulings.

The Voting Rights Act has long received deep bipartisan support, but the new legislation will almost certainly be filibustered with scant or no Republican backing.

As a confrontation on the filibuster approaches, Democrats are no longer talking about scrapping a procedural rule and are instead framing their mission as trying to restore the Senate as a functioning body.

“The reflexive obstruction from Senate Republicans is not — is not — how the Senate is supposed to work,” Mr. Schumer said.

As Mr. Biden suggested on Thursday, Democrats are exploring alternatives such as forcing filibustering senators to hold the floor, which would put more of the onus on those blocking legislation.

“It is not all or nothing,” said Senator Angus King, independent of Maine, who this week said he had come around to the idea of changing the rules if that was what it took to move forward with a voting rights measure.

He and many Democrats also seem to be coming to grips with one of the leading rationales for not changing the filibuster — that once the opposing party regains power, it will force through objectionable legislation that can no longer be blocked using the tactic. Given the stakes, some Democrats say they may just have to accept that possibility.

“Majorities come and go,” Mr. King said. “There is not much doubt that if a significant change is made, it will be used by the other party when and if they are in power.”

But, he said of voting rights, “If we don’t do something about what is going on around the country, then everything else sort of falls by the wayside.”