Kevin M. Warsh moved one step closer on Wednesday to becoming the next chair of the Federal Reserve, after Republicans on a key Senate panel advanced his nomination to the full chamber for a final vote.

The development all but guaranteed that Mr. Warsh, a Fed governor from 2006 to 2011, will eventually take over leadership at the central bank. But it remains unclear whether the Senate can complete the confirmation process before May 15, when the term of Jerome H. Powell, the current chair, expires.

For weeks, Mr. Warsh’s nomination was bogged down while the Trump administration investigated Mr. Powell over cost overruns related to a renovation of Fed headquarters. Mr. Powell, along lawmakers and Fed watchers, saw the inquiry as politically motivated, part of a broader campaign by President Trump to pressure Fed policymakers into lowering interest rates.

That led one key Republican member of the Senate Banking Committee, Senator Thom Tillis of North Carolina, to refuse to allow a vote on Mr. Warsh until federal prosecutors dropped their investigation into Mr. Powell. The government complied last week, and Mr. Tillis soon after released his chokehold on Mr. Warsh, whom the senator has described as highly qualified for the top Fed role.

The 13-11 panel vote on Wednesday saw every Democrat oppose Mr. Warsh. Senator Elizabeth Warren of Massachusetts, the leading Democrat on the banking committee, has repeatedly referred to him as a “sock puppet,” and questioned whether he can appropriately manage the nation’s monetary policy without bending to the influence of the White House.

“A vote today by this committee to advance Mr. Warsh will bring the president one step closer to completing his illegal attempt to seize control of the Fed and to artificially juice the economy,” Ms. Warren said shortly before the vote.

Ms. Warren and other Democrats have also taken issue with Mr. Warsh’s finances. If he is confirmed, he would enter office as perhaps the wealthiest Fed chair in history.

Mr. Warsh has promised to divest more than $100 million in assets. Still, Democrats have criticized Mr. Warsh for both the timing of those sales and the lack of transparency around them.