WASHINGTON — Last month’s appointment of Robert S. Mueller III as a special counsel to investigate possible collusion between the Trump campaign and Russia enraged President Trump. Yet, at least initially, he holstered his Twitter finger and publicly said nothing.

But behind the scenes, the president soon began entertaining the idea of firing Mr. Mueller even as his staff tried to discourage him from something they believed would turn a bad situation into a catastrophe, according to several people with direct knowledge of Mr. Trump’s interactions. A longtime friend, Christopher Ruddy, surfaced the president’s thinking in a television interview Monday night, setting off a frenzied day of speculation that he would go through with it.

For now, the staff has prevailed. “While the president has every right to” fire Mr. Mueller, “he has no intention to do so,” the White House spokeswoman Sarah Huckabee Sanders told reporters late Tuesday.

But people close to Mr. Trump say he is so volatile they cannot be sure that he will not change his mind about Mr. Mueller if he finds out anything to lead him to believe the investigation has been compromised. And his ability to endure a free-ranging investigation, directed by Mr. Mueller, that could raise questions about the legitimacy of his Electoral College victory, the topic that most provokes his rage, will be a critical test for a president who has continued on Twitter and elsewhere to flout the advice of his staff, friends and legal team.