US President Donald Trump is considering terminating special counsel Robert Mueller, who was appointed to lead the FBI investigation into Russia's alleged ties to the 2016 US presidential election.

"I think he's weighing that option," Christopher Ruddy, the CEO of Newsmax Media, told PBS' "NewsHour" on Monday. "I think it's pretty clear by what one of his lawyers said on television recently."

"I personally think it would be a very significant mistake," Ruddy added.

Ruddy, a longtime friend of Trump's who consults with him often, was at the White House on Monday.

Mueller was appointed on May 17 by the Justice Department as special counsel overseeing the investigation into Russia’s alleged meddling in the 2016 US elections and related matters.

Mueller was the FBI director from 2001 to 2013. He was appointed by former President George W. Bush and his original ten-year term was given a two-year extension by former President Barack Obama.

Mueller was succeeded by James Comey, who was fired by Trump last month while he was overseeing the investigation into Russian meddling into the 2016 election and potential collusion with the Trump campaign.

In his Senate testimony last week, Comey said Trump requested loyalty from him and asked that he drop an investigation into former national security adviser Michael Flynn.

Trump, however, lacks the legal authority to directly fire Mueller. US Justice Department regulations outline that a special counsel can be fired by the US attorney general "for cause."

A person familiar with Trump's thinking said Tuesday that it is "unlikely" he will fire Mueller, but acknowledged that it is often difficult to predict Trump's behavior.

A source close to Trump said the president is being counseled to avoid firing the special counsel. "He is being advised by many people not to do it," the source said.

US House of Representatives Speaker Paul Ryan said he would be "surprised" if Trump fired Mueller.

"I think he should let Bob Mueller do his job, do his job independently, and do his job quickly, because I think that that's what he would want to have happen," Ryan told conservative commentator Guy Benson.