A number of aides and attorneys were against the notion, believing that it would create a firestorm of criticism against Trump, the source said.
According to the source, White House counsel Donald McGahn and other attorneys made clear they were not in favor of such a move. The news of Trump’s desire to fire Mueller was first reported by the New York Times.
According to the source, McGahn has considered resigning on at least two occasions. But the source did not know whether those considerations dealt specifically with the potential firing of Mueller.
However, the source was told there were times when McGahn was frustrated about issues in the White House.
The source also pointed out that the president vented about removing Mueller in fits and starts last summer.
In July, Trump apparently became frustrated again and there were discussions, according to the source, about finding conflicts of interest and other negative material on Mueller to weaken his credibility.
The source believes that if the effort found enough material or something weighty enough, the subject of firing Mueller would likely have come up again.
Ultimately, Trump did not execute any order to fire Mueller.
The New York Times reported that Trump “backed down after the White House counsel threatened to resign rather than carry out the directive.”
When reached by ABC News, Peter Carr, the special counsel’s spokesman, declined to comment on the New York Times' story.
Ty Cobb, the White House lawyer leading the response to the investigation, told ABC News, "We decline to comment out of respect for the Office of the Special Counsel and its process."