That slot has remained unfilled so far, partly due to Republican opposition on Capitol Hill. Backers argue that Tarullo would be a perfect fit. Sheila Bair, former chief of the Federal Deposit Insurance Corp., argues that the Dodd-Frank provision creating a Senate-confirmed vice-chairman position in charge of bank supervision is important because it would permanently elevate the status of bank supervision at the central bank. Many have pointed to the 2008 financial crisis as proof that the Fed historically has given short shrift to its bank supervision duties.

"The Fed governors still spend much of their time on monetary policy, yet monetary policy only works if you have a stable banking sector," Bair said.

Whoever the Obama administration nominates for the role would have a difficult time obtaining the filibuster-proof 60 votes to be approved by the Senate for the position. Opponents contend that Tarullo would have a particularly hard time gaining approval.

"They Republicans think he's a big government guy who is too close to Obama," said Brookings Institution fellow Douglas Elliott. "The Republicans have chosen to say that Dodd-Frank didn't do anything and was a sell-out to the banks and blocking a confirmation of Tarullo or any Obama pick in this position fits in that narrative."

It is also unclear whether Tarullo would have the same working relationship with a new Fed chief as he did with Bernanke.

People familiar with Tarullo and Summers, who is a top candidate for the Fed chairman job, point out that they are friends who speak frequently. However, they also note that the two individuals are also strong-willed and might butt heads on approaches to bank supervision as well as responsibilities. Some believe the hands-on Summers would try taking authority away from Tarullo if he can.

Michael Krimminger, former general counsel at the FDIC, said he believes it is likely that Tarullo leaves the central bank if Summers is appointed. "Perhaps they could work together as they interacted in the transition, but both have strong personalities," Krimminger said.

A Fed with Yellen in charge would likely move in lock-step with Tarullo's agenda. Yellen, another leading candidate for the job, previously telegraphed that she supports an approach championed by Tarullo to implement steeper capital surcharges on firms that pose the greatest risk to financial stability. She recently said she is opposed to "blunt approaches" like limits on bank size or the resurrection of Glass-Steagall-style separation of commercial banking from investment banking. Tarullo recently said that restoring the Depression-era law was not high on his list of priorities.

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