The possible loss of Mr. Inhofe’s support means “that Pruitt is in big trouble,” John Feehery, a Republican strategist who worked for former House Speaker Dennis Hastert and the former House majority leader Tom DeLay, wrote in an email.

Ultimately, of course, Mr. Pruitt’s fate depends on President Trump, who until now has continued to back his E.P.A. chief, cheering his aggressive efforts to roll back Obama-era environmental regulations.

Asked about Mr. Pruitt on Monday, the White House press secretary, Sarah Huckabee Sanders, said: “Administrator Pruitt has done a good job of implementing the president’s policies, particularly on deregulation; making the United States less energy-dependent and becoming more energy independent. Those are good things. However, the other things certainly are something that we’re monitoring and looking at and I’ll keep you posted.”

Mr. Inhofe said he had not spoken with Mr. Trump or Mr. Pruitt in recent days, but the president and the Oklahoma senator have a good relationship. They spoke several times by phone during Mr. Trump’s presidential campaign, and Mr. Inhofe’s influence on environmental policy is evident throughout the Trump administration: many of his former staff members now work at the White House and the E.P.A.

Indeed, much of Mr. Trump’s environmental agenda, like undoing E.P.A. climate change rules and pulling the United States out of the Paris climate change accord, is taken straight from the playbook Mr. Inhofe has pushed for years on Capitol Hill.

Privately, many of the former Inhofe staff members now in the Trump administration say they are fed up with the chaos surrounding Mr. Pruitt and are more than ready to see him go. They noted that another former member of Inhofe’s staff, Andrew Wheeler, had recently been confirmed as Mr. Pruitt’s deputy, and that Mr. Wheeler, a former coal lobbyist, was well positioned to continue the agenda of rolling back environmental rules should Mr. Pruitt step down.