President Trump’s persistent demand for the elimination of the Senate filibusters didn’t garner any public support from the Republicans who run the chamber, but behind the scenes, the president’s allies are starting to argue that it isn’t a crazy idea or institutional heresy.

The case is being made — in public by Mr. Trump and in private by like-minded conservative leaders — that the polarization of the two parties, especially with the purge of moderates from the ranks of Democratic lawmakers, has rendered the 60-vote filibuster test an obsolete tradition that has become harmful to the democratic process.

The Senate is no longer a stately cooling saucer for overheated legislation but a stagnant swamp where bills putrefy, the argument goes.

Because of the requirement for 60 votes to advance most legislation in the Senate, the fate of Mr. Trump’s top priority to build a wall on the southeastern border now rests in the hands of the minority party.

Senate Democrats’ lockstep opposition to the wall is enough to derail a funding bill, potentially provoking a government shutdown over the issue despite support from the White House and majorities in the House and Senate.

It also created a roadblock for more than 300 bills passed by the Republican-run House, including measures such as Kate’s Law that would increase penalties for deported illegal immigrants who return to the U.S., a rewrite of the Dodd-Frank financial law and numerous bills to combat human trafficking.

“The president, quite honestly, is right on the filibuster,” said Jim McLaughlin, a pro-Trump Republican strategist and pollster. “There are just no moderate Democrats left. They just don’t exist anymore.”

He cited Sen. Joe Manchin III, a West Virginia Democrat heralded as the party’s most moderate voice. But even Mr. Manchin refused to break the Democrats’ united front against the repeal and replacement of Obamacare, he said.

Mr. Trump repeatedly took to Twitter last week to pressure Senate Majority Leader Mitch McConnell, Kentucky Republican, on changing the chamber’s filibuster rules. He has been calling strongly calling for it since the repeal-and-replace bill died last month in a 49-51 vote, one short of the votes needed to pass under special rules for budget measure — a process called “budget reconciliation.”

“If Senate Republicans don’t get rid of the Filibuster Rule and go to a 51% majority, few bills will be passed. 8 Dems control the Senate!” the president wrote in one of the tweets.

Mr. McConnell has scoffed at the idea. Like other critics of the president, he has said that ending the filibuster wouldn’t have saved the Obamacare repeal, which was not subject to a filibuster under reconciliation.

“It’s pretty obvious on health care our problem was not the Democrats,” he said recently. “The votes were simply not there.”

Indeed, Senate Republicans with their thin 52-seat majority couldn’t muster the votes with no support from Democrats and lost three senators from their party: John McCain of Arizona, Susan M. Collins of Maine and Lisa Murkowski of Alaska.

Last week, however, Mr. Trump explained that the filibuster threat forced Republicans to put up a less-attractive bill in order to meet the strict requirement of reconciliation, which does not allow policy measures on a vehicle that passes with a 51-vote majority.

“It’s a trick. We needed 51 votes,” Mr. Trump said at a campaign-style rally in Phoenix. “But some of the best things in health care require 60 votes.”

He said that some of the more popular measures to replace Obamacare, such as allowing people to purchase insurance plans across state lines, which Mr. Trump has long advocated, could not be included in the bill under reconciliation rules.

He said the bill Republicans wanted to pass would have easily received 51 votes in the Senate, but a Democratic filibuster prevented that bill from seeing the light of day.

“You got to get rid of the filibuster rule. You’ve got to go to a majority. You’ve got to go to 51 votes, and if they don’t do that, they’re just wasting time,” said Mr. Trump.

Blowing up the rules frightens Senate Republicans, who fear being blamed for altering the nature of the chamber to more closely resemble the impetuous, majority-ruled House.

What’s more, they fear Democrats ramming though their agenda when their roles are reserved.

But keeping the filibuster now doesn’t preclude Democrats from revoking it later, as Mr. Trump has warned.

“If Republican Senate doesn’t get rid of the Filibuster Rule & go to a simple majority, which the Dems would do, they are just wasting time!” he tweeted.

Senate Democrats, when they ran the chamber with Sen. Harry Reid as majority leader, were the first to trigger the “nuclear option” in 2013 to end the filibuster of most executive nominations. Mr. Reid called it necessary to break through Republican obstruction of President Obama’s nominations.

The move gave Mr. McConnell and the Republican majority cover this year to again go nuclear to end filibusters of Supreme Court nominees. Without the rule change, Justice Neil M. Gorsuch would never have been elevated to the high court, which remains Mr. Trump’s crowing achievement in the eyes of most Republicans.

The complete obliteration of Senate filibusters for many stands as a bridge too far.

“I’m inclined, much as I see its limitations, to say leave the filibuster alone,” said Boston University professor Michael T. Corgan, a scholar of the presidency and Congress.

He said the filibuster has worked in the Senate for more than 100 years. The first filibuster was mounted over the U.S. expedition to liberate Cuba in 1870s.

“I don’t like to see procedural changes to suit the needs of the moment or the present incumbent powers in Congress,” Mr. Corgan said. “Filibusters slow things down in the legislature, and that’s usually a good thing. Bills passed in haste or in the heat of the moment’s passions usually end up badly.”

Most important, he said, the filibuster supports the vision of James Madison and other Founding Fathers who wanted to protect policy or political minorities.

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