Robert Mueller has come under attacks recently, with the Speaker of the House 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".