“I asked for their perspective,” Mr. Chaffetz said. “They were overall complimentary of the process” of moving nominees through the ethical vetting, he said. However, he added, “I think there are a lot of concerns about Mr. Shaub.”

The scrutiny of the ethics agency is the latest deviation from a generation of standard practices as the new administration prepares to take over. The tone has been largely set by Mr. Trump, who, unlike a long list of past presidents, has declined to release his tax returns.

But Mr. Shaub has made some unusual moves himself. First, his office posted a puzzling series of messages on Twitter in late November applauding Mr. Trump for divesting his assets — something he has not done. The tone was not unlike the distinctive flavor of Mr. Trump’s own messages on Twitter. Then, on Wednesday, Mr. Shaub held a news conference at the Brookings Institution to criticize Mr. Trump’s ethics plan. The president-elect intends to retain his stake in his business empire and put the holdings into a trust controlled by his adult sons, a plan that other ethics experts have also said is insufficient.

“I don’t think it was particularly wise to do,” said Paul C. Light, a professor of public service at New York University who worked on Capitol Hill as an adviser on transitions. “It is a moment fraught with a great deal of controversy and partisan criticism, and I think it is best if you are in an investigatory organization to hold your fire, do your job and do the best you can. It’s just not the time for it.”

Before turning their attention to Mr. Shaub’s office, Republicans tried, in one of their first acts in the new Congress, to gut the independent House ethics office. The effort was ultimately halted amid a political backlash.

The moves have left Democrats aghast. “First, House Republicans tried to gut the Office of Congressional Ethics,” Senator Chuck Schumer of New York, the Democratic leader, said in a statement. “Now they’re trying to handcuff the Office of Government Ethics. Mr. Chaffetz’s attempts to bully Mr. Shaub out of doing his job are absolutely despicable.”