There are signs that lawmakers are open to considering the ideas. Representative Trey Gowdy of South Carolina, the new Republican chairman of the House Oversight and Government Reform Committee, said he was preparing to meet with Mr. Shaub. The effort could be a test of what kind of appetite Mr. Gowdy has to challenge the Trump administration as the chairman of what is traditionally the most active oversight committee in Congress.

Representative Elijah E. Cummings of Maryland, the committee’s top-ranking Democrat, also wants to discuss the ethics office and ways to strengthen it.

“I look forward to having a productive conversation with Mr. Shaub and Elijah Cummings before the outgoing director leaves office,” Mr. Gowdy said in a statement in response to questions from The New York Times about possible changes in the authority granted to the office, known as O.G.E. “The discussion will include ways to improve the ethics process and instill confidence in O.G.E.”

Mr. Cummings is already drafting legislation with the hope of gaining Mr. Gowdy’s support, members of his staff said. It will incorporate some of Mr. Shaub’s proposals, albeit the less contentious provisions that stand a chance of passing in a Republican-controlled Congress.

“The Office of Government Ethics has an impossible job under this administration because President Trump has ignored its advice, undermined its authority and openly flouted ethics rules,” Mr. Cummings said in a statement. “Now more than ever, it is important for Congress to act to strengthen O.G.E. and protect its independence.”

Mr. Shaub’s relationship with Mr. Trump has been tense since even before Inauguration Day. He publicly pushed Mr. Trump to do as every other president in recent decades has voluntarily done and sell his assets before taking office to avoid conflicts of interest.