Of Mr. Tiberi, Mr. Cole said that he was “happy for my friend, but Congress will be a poorer place without him.”
A series of long-tenured Republicans have announced plans to leave Congress in recent months, including Senator Bob Corker of Tennessee and some of Mr. Tiberi’s House colleagues, including Charlie Dent of Pennsylvania, Dave Reichert of Washington State and Ileana Ros-Lehtinen of Florida. In most cases, they have voiced exasperation with dysfunction in Washington and their party’s inability to deliver major legislative victories, at times fuming at Mr. Trump for undermining policy making in Congress.
Mr. Tiberi, however, would be a particularly striking congressional refugee. He was a close ally of John A. Boehner, the former House speaker, and had considered challenging Senator Sherrod Brown, Democrat of Ohio, next year. Mr. Tiberi, who was first elected in 2000, had more than $6.6 million in his re-election account as of the start of the month.
Despite losing a contest for the chairmanship of the Ways and Means Committee — the job went to Representative Kevin Brady, Republican of Texas — Mr. Tiberi has remained a senior member of the powerful tax-writing panel.
A sudden resignation would prompt a special election for Mr. Tiberi’s seat, in a solidly Republican district outside of Columbus, the state capital.
Several other Republicans have headed for the exit since the summer, including Representative Dave Trott of Michigan, a junior lawmaker who expressed disgust at the state of play in Washington, and Tim Murphy of Pennsylvania, who resigned this month after it was revealed that he had an extramarital affair with a woman whom he urged to have an abortion.