Republican control of the Senate makes it likely that Alabama's Richard Shelby, the senior Republican on the Committee on Banking, Housing, and Urban Affairs, will become its chair, steering the committee that oversees the Fed.
"I don't think he's a big fan of the Fed," Hugh Johnson, chairman of Hugh Johnson Advisors, said of Shelby, a onetime central bank backer who turned into a Fed critic during the financial crisis.
Shelby opposed Yellen's nomination as vice chair in 2010 and then as chair last year. He also fought President Barack Obama's 2010 nomination of Nobel Prize-winning economist Peter Diamond to serve as Fed governor, with Diamond subsequently withdrawing from consideration. Two of seven board seats remain vacant.
"Anyone very dovish is not even going to get a hearing or a vote," said Mark Calabria, a former staff member for Shelby and now director of financial regulation studies at the Cato Institute, a libertarian think tank.
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"Shelby, who is expected to chair the banking committee, is sympathetic to getting the Fed out of bank regulation, but getting 60 votes for that is a tough thing to do," Calabria added.
Legislation that passed the House in September and sponsored by Georgia Republican Paul Broun would require the comptroller general to audit the Fed's board of governors and banks within a year and report back to Congress on its findings. Rand Paul, R-Kentucky, has introduced companion legislation in the Senate.
"Rand Paul will likely come down and demand a Fed audit, but it's an open question at this point whether there will be a vote. There's a small handful of Republicans who were not terribly sympathetic to it," Calabria said.
The measure has a good chance of passing should it be brought to a vote, Calabria said.
House Financial Services Committee Chairman Jeb Hensarling, a Texas Republican, told Bloomberg News in an October statement that he had already "held a number of conversations with Sen. Shelby, who might, and hopefully will, be the next Senate Banking Committee chairman, and we have had preliminary discussions about what he would like to achieve."
But it remains to be seen whether the Senate would actually take up the House's efforts to reduce Fed authority.
"There have been some pretty intense periods of political backlash, and this is among the more intense," said Kohn. He voiced optimism that the Senate "will have a more reasoned" approach in considering the role of the central bank.
"Republicans have been vocal about wanting to tighten the reins on the Fed, and will probably make an attempt to do that, but it won't happen until all the new members are in place next year," said Randy Frederick, managing director of trading and derivatives at Charles Schwab.
The idea of curbing the central bank's authority is a topic of discussion on Wall Street, and Frederick and others generally oppose the idea of limiting the Fed's power or independence regardless of their political affiliation. Supporters of the central bank credit it with rescuing the U.S. and global economy during the financial crisis, while its critics often deride it for easy money policies they fear will create a bubble and the next financial crisis.
"I personally think it would be a disaster to put guard rails in place that would not allow the Fed to do its job," said Russell, who added that the U.S. economy's recovery in recent years is "by and large a construct of the Fed's actions."
"Any idea of insisting the Fed detail its exit strategy to me is troubling, but I don't think that that will get far, because it will be greeted by a veto by Obama," said Johnson.
And, while Obama would surely veto any attempt to curtail the central bank's independence, the Fed and Yellen are likely to encounter pushback should they opt to slow the central bank's departure from its record stimulus.
"If the markets roll over and everyone says the Fed needs to do another round of quantitative easing, that would get politically difficult. That's where Republicans have more of a problem is in the unconventional space," said Peter Boockvar, chief market analyst at The Lindsey Group.
Conversely, Calabria believes once the Fed starts raising interest rates, the central bank will face an outcry from the left, which he predicts will argue the labor market is not yet doing well enough for Fed tightening.
As Calabria put it: "To continue an exit strategy, Republicans will be more her friends than Democrats. The best thing to happen to Yellen is Republicans taking the Senate."