Conservative allies are adding pressure on the special counsel overseeing the investigation into alleged Russian interference in last year's election.
Robert Mueller has come under attack recently, with the former House Speaker casting doubts over his credibility.
Trump advocate Newt Gingrich urged the president to "rethink" Mr Mueller's position.
A close friend said Mr Trump saw firing him as an option, but the White House said both never discussed this issue.
Mr Mueller, who was tasked by Deputy Attorney General Rod Rosenstein with overseeing the justice department's investigation, has started selecting members of his team, many of them criminal law experts.
If confirmed, his dismissal could have explosive implications and renew criticism of Mr Trump, who has been under intense pressure following his firing of James Comey as FBI director, and accusations that he might have tried to obstruct the investigation.
The very appointment of Mr Mueller only happened after calls for such a move sparked by Mr Comey's firing. Mr Trump considers the inquiry a "witch hunt".
Troubles for Trump? By Anthony Zurcher, BBC North America reporter
The latest appointments from Robert Mueller's special counsel investigation could be a more ominous indication of trouble on the horizon for the Trump administration.
The hires help shed light on the direction of the probe and the seriousness with which Mr Mueller is taking the enterprise.
Donald Trump has called the ongoing investigation a "witch hunt", a "hoax" and an excuse by Democrats for why they lost the presidential race. Mr Mueller, however, is assembling a team built for the long haul, with the talent and experience to take cases to trial and, if necessary, send people to prison.
So far, Mr Trump has only directed his criticism at former FBI Director James Comey. It may only be a matter of time before some of that attention is directed at Mr Mueller, however.
Back in 1990, Bill Clinton and Democrats succeeded in turning independent counsel Ken Starr into a villain for much of the public and painting his investigation into the president's relationship with Monica Lewinsky as a tawdry political endeavour.
Trump supporters may try to do the same now, although Mr Mueller's sterling reputation and Mr Trump's sinking public approval makes that a difficult task.
Mr Mueller's appointment was initially praised by both Republicans and Democrats, but some influential conservatives have intensified their attacks, openly defending his dismissal.
Prominent radio host Mark Levin wrote on Facebook that "Mueller must step aside", while commentator Ann Coulter said there was no point in keeping him after Mr Comey confirmed that Mr Trump was not under investigation. She said on Twitter: "Why do we need a special counsel now?"
Meanwhile, Mr Gingrich, also on Twitter, said: "Republicans are delusional if they think the special counsel is going to be fair. Look who he is hiring."
Mr Trump does not have the authority to dismiss Mr Mueller, and he would have to ask Mr Rosenstein, who appointed the special counsel, to do so.
This would evoke memories of the "Saturday Night Massacre" of 1973, when President Richard Nixon sought to dismiss a special prosecutor.
But a long-time friend of Mr Trump said the president was considering sacking Mr Mueller.
Christopher Ruddy, chief executive of conservative group Newsmax Media, Mr Ruddy was at the White House on Monday, but US media reported that he did not meet Mr Trump while there.
In an interview with PBS Newshour, he said: "I think he is considering, perhaps, terminating the special counsel."
"I think he's weighing that option," he added, saying there were concerns about conflicts of interest, including Mr Mueller's interview to replace Mr Comey at the FBI and his law firm, which represents members of Trump family.
But he said: "I personally think it would be a very significant mistake."
Hours after the interview, White House spokesman Sean Spicer did not deny the claim, but said: "Mr Ruddy never spoke to the president regarding this issue. With respect to this subject, only the president or his attorneys are authorised to comment."