Newsmax Media CEO Christopher Ruddy made the comments on the eve of testimony at the Senate Intelligence Committee by US Attorney General Jeff Sessions over the Trump-Russia probe.
His comments came just days after explosive comments by ousted FBI director James Comey to the same committee, suggesting that Trump sought to interfere with the investigation.
"I think he's considering perhaps terminating the special counsel. I think he's weighing that option," Ruddy told PBS "NewsHour" when asked if Trump would let Mueller continue in his role in the Russia probe.
"I personally think it would be a very significant mistake," Ruddy added.
A White House official stressed that Ruddy "speaks for himself," not the Trump administration.
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CNN cited a source close to Trump as saying the president was "being advised by many people" to not fire Mueller, who led the FBI between 2001 and 2013.
The top Democrat on the House Intelligence Committee, Adam Schiff, warned on Twitter that "If President fired Bob Mueller, Congress would immediately re-establish independent counsel and appoint Bob Mueller. Don't waste our time."
The president sacked Comey in early May. Given that as FBI director Comey was overseeing the probe into Russia, and its possible collusion with the Trump team, the firing has led to questions about potential obstruction of justice.
Sessions, who recommended in a signed memo that Comey be fired, may end up claiming executive privilege as a means of limiting the scope of his testimony.
Trump has expressed frustrations with Sessions, one of the president's earliest high-profile backers.
