Amid the flurry of Democrats’ questioning whether President Biden should or will remain his party’s presidential nominee, former President Donald J. Trump has stayed unusually quiet on the issue publicly.

Mr. Trump, rarely one to shy away from sharing his opinion, has not been fully silent since last week’s debate, giving a handful of radio interviews and keeping up a steady stream of posts and videos on his social media platform, Truth Social. But Mr. Trump has largely sat back and allowed the Democratic Party to dominate the debate over Mr. Biden’s political future, in a signal of his preferred opponent.

After months of relentlessly attacking Mr. Biden as too physically and mentally weak to lead the country, the former president has been content to let the news coverage of Democrats’ doubting their party’s leader take hold, according to two advisers, who spoke on the condition of anonymity to discuss strategy.

His relative lack of public comments on the issue also to some extent reflects his desire for Mr. Biden to stay in the race and his confidence that he can easily beat the president in November, one of the advisers said.

A New York Times/Siena College poll conducted after the debate and released on Wednesday suggested that some Republican voters agreed: 28 percent of them said they thought Mr. Biden should remain the Democratic nominee, an uptick from 21 percent in a poll conducted before the debate.

On Monday, Mr. Trump publicly dismissed the idea that the president would be replaced on the Democratic ticket.

“If you listen to the professionals that do this stuff, they say it’s very hard for anybody else to come into the race,” Mr. Trump said in an interview with John Reid, a Virginia-based talk radio host.

And in an echo of a talking point that Mr. Biden’s Democratic allies have long wielded to argue that he is best positioned to beat the former president, Mr. Trump has also argued that polling showed that “Biden does better than the people they’re talking about using to replace him.”

The day after the debate, he argued at a rally in Virginia that Mr. Biden polled better in head-to-head matchups against him than did either Vice President Kamala Harris, whom he said he would “be very happy” to run against, or Michelle Obama, the former first lady.

Two polls released on Tuesday somewhat deflated that claim: A CNN poll found that Ms. Harris polled two percentage points ahead of Mr. Biden in a hypothetical contest against Mr. Trump, though he still defeated her. And an Ipsos/Reuters poll found that Mrs. Obama — a long-shot potential option for Democrats given that she has repeatedly said she has no interest in running — beat Mr. Trump, 50 percent to 39 percent, in a hypothetical matchup.

If Mr. Biden stepped aside, Mr. Trump would lose two lines of attack that have been central to his campaign. He has spent years attacking Mr. Biden as “sleepy,” posting videos of Mr. Biden’s stumbles, mocking his speech and performing cartoonish imitations of him, attacks that he could not easily deploy against another opponent.

And for the last several months, Mr. Trump has tried to appeal to undecided voters by directly comparing his time in office with Mr. Biden’s, often in misleading terms. That message would be hamstrung if another candidate replaced Mr. Biden on the ticket.

A new opponent could open up new political challenges. Mr. Trump could face a younger opponent who could appeal to voters worried about both candidates’ ages and looking for fresh alternatives to two men who each have had a shot at a White House term.

“I don’t think anybody in the Trump campaign has ever said they want Biden off the ticket,” said Corey Lewandowski, a longtime Trump adviser who is now an adviser for the Republican Party’s nominating convention. He added that the matchup of “two candidates who America both knows very well and has a record to compare is one that favors us very, very much.”

The Heritage Foundation, a major conservative group, has also been exploring possible legal challenges it could mount that would make it difficult to replace Mr. Biden on the ballot in some states if he withdrew.

Mike Howell, the executive director of Heritage’s Oversight Project, said the group was eyeing key battleground states like Georgia, Nevada and Wisconsin where laws might make it difficult to put a different Democrat on the ballot.

In a statement on Wednesday, Mr. Trump’s two campaign managers, Susie Wiles and Chris LaCivita, expressed their confidence that Mr. Trump could “beat any Democrat” in November. They accused Democrats now turning on Mr. Biden of being hypocrites, saying that “every one of them has lied about Joe Biden’s cognitive state and supported his disastrous policies over the past four years.”

There have been signs that some people in Mr. Trump’s orbit are preparing more seriously for the possibility, however distant, of a matchup against another Democratic candidate this fall. The Trump campaign and Republican allies have ramped up attacks on Ms. Harris, who has long been a target of the right.

In their statement on Wednesday, Mr. Trump’s campaign managers called her “Cackling Copilot Kamala Harris,” both mocking her mannerisms and directly linking her to Mr. Biden’s policies. During the debate, the campaign ran an ad suggesting that Mr. Biden was incapable of leading the country through a second term and warning that Ms. Harris was waiting in the wings to take over.

On Tuesday morning, Make America Great Again Inc., the leading super PAC supporting Mr. Trump’s presidential campaign, sent out a list of attacks on Ms. Harris that essentially argued that she would be no better than Mr. Biden, particularly on immigration, an issue Mr. Trump has made central to his campaign.

On Wednesday, the campaign committee for House Republicans announced a new digital ad that linked Ms. Harris to Mr. Biden’s border policies. “Vote Republican. Stop Kamala,” a title card at the end of the ad reads.

“Every good campaign looks at every possible contingency,” Mr. Lewandowski said. “The campaign’s strategy isn’t shifting, but it would be a dereliction of duty to not be prepared should Joe Biden drop out of the race.”

Senator Lindsey Graham, Republican of South Carolina and a close Trump ally, issued a warning of sorts on social media about how the 2024 race might shift if Ms. Harris became the nominee.

“I believe the Trump Campaign realizes the 2024 race could very soon dramatically shift away from Biden’s capabilities to a fight for the heart and soul of the country,” Mr. Graham wrote on X.

And if the scenario played out, Mr. Graham added, Republicans would need to “build on President Trump’s ability to expand the demographic reach of our party in 2024.”