Administration officials earlier this year settled on Marvin Goodfriend, a Carnegie Mellon economist, for a second seat, but Mr. Trump has not made the nomination.

As for Ms. Yellen’s position, an administration official, speaking on condition of anonymity, said Wednesday that the president was still reviewing candidates. Gary Cohn, the chief White House economic adviser, has been mentioned as a possible nominee. But a person close to the president said Mr. Cohn had fallen out of favor — although not totally out of the running — after expressing distress last month in an interview with The Financial Times over the administration’s response to violent clashes in Charlottesville, Va.

The Fed’s board had at least five members from 1936 until 2009. The head count was reduced to four during brief stretches in 2009, 2010, and 2014. But by far the longest stretch in Fed history with only four board members began in March of this year.

It is the first time that a majority of votes on the Federal Open Market Committee, which makes monetary policy, have been held by the presidents of the Fed’s regional reserve banks — who, unlike most officials with policy authority, are neither directly elected nor confirmed by the Senate. The committee includes the members of the board and the 12 reserve bank presidents, five of whom hold votes at any given time.

Mr. Fischer spent two decades as a professor at the Massachusetts Institute of Technology, where his students included Mario Draghi, now president of the European Central Bank, and Ben S. Bernanke, Ms. Yellen’s predecessor as Fed chief.

Then, in the late 1980s, he embarked on a second career in public service, first as chief economist at the World Bank, then as a senior official at the International Monetary Fund and as governor of the Bank of Israel. By the time he arrived at the Fed, he was widely regarded as leading expert on both the theory and the practice of central banking.

Analysts said Mr. Fischer’s departure was unlikely to shift monetary policy in the near term. The Fed is widely expected to announce after its next meeting in mid-September that it will begin to reduce its holdings of Treasuries and mortgage-backed securities.

The Fed has raised its benchmark interest rate twice this year, to a range between 1 percent and 1.25 percent. Most officials predicted in June that the Fed would raise rates one more time this year, but sluggish inflation could cause the Fed to hesitate.