Last week, Representative Trey Gowdy, the South Carolina Republican who is chairman of the House Oversight Committee, opened an investigation into Mr. Pruitt’s expenditures on first-class travel, which the E.P.A. has said were necessary because passengers have made threatening remarks to Mr. Pruitt during flights. Mr. Gowdy, who gained prominence for his investigations into Hillary Clinton over the 2012 attacks in Benghazi, Libya, that left four Americans dead, on Sunday had pointed words for Mr. Pruitt.

"I would be shocked if that many people knew who Scott Pruitt was,” Mr. Gowdy said on Fox News. “So the notion that I have to fly first class because I don’t want people to be mean to me, you need to go into another line of work if you don’t want people to be mean to you, like maybe a monk where you don’t come in contact with anyone.”

Privately, a number of Republican staffers and strategists say that last week’s Senate confirmation of Mr. Pruitt’s deputy, Andrew Wheeler — a former coal lobbyist who worked for years on Capitol Hill — would make Mr. Pruitt’s departure easier because Mr. Wheeler would be in place as acting E.P.A. chief and could be relied upon to implement an agenda of rolling back environmental regulations.

Some Republican lawmakers have also pointed out the similarities between Mr. Pruitt’s case and that of Tom Price, Mr. Trump’s former secretary of health and human services. Mr. Price was forced to resign after racking up at least $400,000 in travel bills for chartered flights.