WASHINGTON — Senior White House staff members are encouraging President Trump to fire Scott Pruitt, his embattled Environmental Protection Agency chief, according to two top administration officials. While Mr. Trump has until now championed Mr. Pruitt, the officials say the president’s enthusiasm may be cooling because of the ongoing cascade of alleged ethical and legal missteps.

Over the past few months, as Mr. Pruitt’s problems have mounted — he is now the subject of at least 11 federal investigations and some Republicans have called for his resignation — Mr. Trump has continued to support his E.P.A. chief on Twitter and in public and private remarks.

But that is likely to change in the coming weeks, the two officials said.

Since last month’s confirmation of Mr. Pruitt’s deputy, the former coal lobbyist Andrew Wheeler, White House staff members say they believe that if Mr. Pruitt is fired or resigns, Mr. Wheeler will continue to effectively push through Mr. Trump’s agenda to help the coal industry and roll back environmental regulations.

Some Republicans have said that Mr. Wheeler, a former Capitol Hill and E.P.A. staff member — known as a low-key but highly experienced Washington insider — would quite likely be as effective, and possibly more so, than Mr. Pruitt at undoing regulations, without drawing the embarrassing headlines of his boss.