John F. Kelly, the White House chief of staff, told President Trump last week that Scott Pruitt, the embattled administrator of the Environmental Protection Agency, needed to go following damaging allegations about ethical infractions and spending irregularities, according to two officials briefed about the conversation.

But Mr. Trump, who is personally fond of Mr. Pruitt and sees him as a crucial ally in his effort to roll back environmental rules, has resisted firing him, disregarding warnings that the drumbeat of negative headlines about the administrator has grown unsustainable and that more embarrassing revelations could surface.

White House officials said Friday that Mr. Trump continues to believe that Mr. Pruitt has been effective in his role, and they stressed that it was up to the president alone to decide his fate.

“No one other than the president has the authority to hire and fire,” Sarah Huckabee Sanders, the White House press secretary, told reporters. “The president feels that the administrator has done a good job at E.P.A.”