Mr. Pruitt also requested that the agency order other items — including leather-bound notebooks, fountain pens and stationery — from which he wanted to omit the E.P.A. seal and upon which he wanted to feature his name prominently, according to Mr. Slotkin and the person who participated in the discussions about the seal . Ultimately, the items retained a small version of the seal, according to several people familiar with the orders.

The debate over souvenirs came as Mr. Pruitt was engaged in personal and public spending that has since become the subject of scrutiny, threatening his tenure at the E.P.A.

Mr. Pruitt has been under fire for renting a condominium for $50 a night from the wife of a lobbyist with business before his agency, as well as for his spending of taxpayer dollars on first-class travel, which he has asserted was necessary for security reasons.

In an interview with The Washington Examiner this month, Mr. Pruitt said he was under attack because he has been effective in enacting President Trump’s regulatory overhaul agenda and opponents would like to stop him. “And do I think that they will resort to anything to achieve that?” he said. “Yes.”

Mr. Trump defended Mr. Pruitt in a weekend Twitter message: “While Security spending was somewhat more than his predecessor, Scott Pruitt has received death threats because of his bold actions at E.P.A. Record clean Air & Water while saving U.S.A. Billions of Dollars. Rent was about market rate, travel expenses O.K. Scott is doing a great job!”

Some critics of Mr. Pruitt’s coin proposal said it missed the point of the gift item. Scott H. Amey, general counsel of the Project on Government Oversight, a watchdog group, said that the coins were intended to honor jobs well done and lift morale. “The coin should reference the E.P.A., and not become tribute to Pruitt or Oklahoma,” he said.