Mr. Trump did not make a formal offer to Mr. Perry when the two men met on Monday. But the people said the president has grown impatient with the department’s current secretary, David Shulkin, and may want to replace him with someone already in his cabinet.

It was unclear if Mr. Perry, who was an Air Force pilot before entering politics, would accept the change in position if Mr. Trump offered it, or if Mr. Trump had a successor in mind to lead the Energy Department