WASHINGTON — Senators will vet Judy Shelton, a prominent Federal Reserve critic, for one of the central bank’s top jobs on Thursday, a moment that many economists will be watching closely because they see her as a potential threat to the Fed’s prized independence.

Ms. Shelton has rhetorically questioned whether America even needs the Fed. She is a longtime fan of pegging the value of the dollar to some common anchor, like gold, a stance that the United States abandoned decades ago and one that would fundamentally change the nature of the central bank. Despite those unconventional views, she is one of President Trump’s two candidates to fill the last remaining seats on the Fed’s seven-member Board of Governors in Washington.

It is not just her heterodox economic views that are making former central bankers and top economists wary, according to interviews. It is the possibility, based on her record, that Ms. Shelton could bend her positions to suit Mr. Trump’s preferences, subordinating monetary policy to the White House’s desires.

The stakes are high. While individual Fed governors have just one vote out of 12 on monetary policy, she could be a natural pick to be the central bank’s next chair should Mr. Trump win re-election. That would put her in a position to change the fabric of the institution. While Fed chairs must build consensus among their colleagues, they have historically been the central players in American monetary policy.