Senator Chuck Schumer of New York, the minority leader, said Tuesday on the Senate floor, “This new hearing schedule is a very good first step, but we still have a ways to go.”

Mr. Trump’s aides have been pressuring the ethics office to expedite its processes, even though it did not get the necessary paperwork from most of his nominees until recently, putting Republicans in a difficult position of either pushing forward with hearings without proper vetting — something most of the chairmen are loath to do — or delaying them in some cases indefinitely.

“Trump is the latest president-elect to begin the process in contemporary history,” said Paul C. Light, a professor of Public Service at New York University who worked on Capitol Hill as an adviser on transitions. “This puts the Republicans under pressure and sets up possible problems down the road. Mr. Trump is asking the Senate to make up for his own failures here.”

The conservative organization America Rising PAC took aim at Mr. Shaub on Monday, highlighting his “history as a Democrat and the double standard he employed as head” of the agency, highlighting his political donations to President Obama as well as his office’s “utter incompetence” in policing financial matters related to Hillary and Bill Clinton. The PAC chided Mr. Shaub for wading into the politics of the transitions, saying his public comments and criticisms had become fodder for “Congressional Democrats to try and score points.”

The PAC noted an informal — and ill-advised — series of postings on Twitter from the office’s official account congratulating Mr. Trump on divesting from his real estate business — a move he had not made.

“The head of President Obama’s ethics department has absolutely zero credibility to criticize others,” Scott Sloofman, a spokesman for America Rising PAC, said in an email, adding that Mr. Shaub’s “outburst over the weekend reeks of partisan politics from an embittered Democrat still reeling from November’s election result.”