Bernanke also opened the door to speculation about his departure during the press briefing after last month's Fed meeting. In response to questions, he did say he has spoken to President Obama but did not offer any specifics on whether he would like to stay on at the Fed. "I don't think that I'm the only person in the world who can manage the exit," he said, a reference to the unwinding of the Fed's current easing policies.
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Within the Obama administration, it is widely assumed that Bernanke wants to depart at the end of his term. But the Fed will not comment further beyond Bernanke's public statements.
"I think he never really signaled that he'd stay," said Mesirow Financial Chief Economist Diane Swonk. She and others say the likely successor is Yellen, who was named to the Fed by President Bill Clinton in 1994.
"I think she's the No. 1 choice. She's the one that's got the most continuity. She's got a great resume that supports it as well. She's not necessarily more dovish than Bernanke," Swonk said. Yellen, 66, famously swayed former Fed Chairman Alan Greenspan on the idea of an inflation buffer—that a low rate of inflation is a good thing, rather than zero inflation, Swonk said.
Yellen, who joined the Fed as president of the San Francisco branch, came from the University of California at Berkeley. She has been seen as a key part of the Fed's dovish core, together with Bernanke and New York Fed President William Dudley.
No major policy changes are expected at this week's two-day meeting, which ends Wednesday.
Other names that have surfaced as candidates include former Treasury Secretary Timothy Geithner, but he is not expected to want the job, and former Treasury Secretary Larry Summers, who is said to be interested in the chairmanship. Former Fed officials Donald Kohn; TIAA-CREF CEO Roger Ferguson; and Princeton economist Alan Blinder are all also viewed as potential candidates. Two current Fed possibilities are Dudley and Fed Gov. Jeremy Stein, but Yellen is viewed as the clear front-runner.
While many on Wall Street believe a transition is coming, there is concern that there could be uncertainty about policy direction, and that could ruffle markets.
"I think people would feel a little more secure, and it's not just domestic investors but internationally. They'd feel a little more secure if the architect of the policy saw it through," said Chris Rupkey, chief financial economist at Bank of Tokyo-Mitsubishi.
Bernanke's retirement may become official by around the time of Jackson Hole in late August. "The insiders wouldn't be saying he's leaving if Bernanke is telling them otherwise. It's not a public announcement but it's seeping out. One important thing about the current policy statement and the policy that'll make the transition easier if it happens is that the Fed is operating on a rules-based approach. There won't be a change in policy there," said Tony Crescenzi, Pimco strategist and portfolio manager.
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Yellen's critics say she may be too willing to let inflation climb while aiming policy at employment. She also would be taking the helm of the Fed when policies that were pursued by Bernanke are put to the test, as the massive amounts of Fed liquidity are unwound and interest rates rise. The Fed's balance sheet has ballooned to about $3.2 trillion, and the Fed continues to purchase $85 billion in mortgages and Treasury securities each month.
"I don't think the market's prepared for this. I think this could be one of the events where we think it's discounted but it's not. This is a major deal, handing over the reins. It's only happened twice in the last 30 years," Rupkey said.
"Yellen has raised her visibility certainly. She's doing a lot more papers. ... But I'm not sure she's on the radar for investors at large. I think nobody is really focusing on this right now. But to the chairman's credit, he has kind of set up with Vice Chair Yellen as head of the communications subcommittee, they've set up something where policy really is on autopilot."
Bernanke also has his critics, but many believe his strong background on the Great Depression made him especially well-suited to serve as Fed chairman during the financial crisis. Bernanke has overseen an increased openness in communications from the Fed, and the markets are now watching for the Fed's targeted inflation rate of 2 percent and unemployment at 6.5 percent for signals when the Fed might stop easing.
"Bernanke's been decisive when other policy makers have been inept, indecisive and dysfunctional. He's been a source of stability and confidence to the extent the markets have built up confidence and are more stable because of his decisiveness and creative actions," said Crescenzi.
(Read More: If I Were 'Dictator,' QE Would End Now, Fed's Lacker Says)
Bernanke came to the Fed chairmanship from the White House, where he was chairman of President George W. Bush's Council of Economic Advisors until he became Fed chairman, when former Fed Chairman Alan Greenspan retired in 2006.
He previously was a Fed governor, and the chair of the Princeton economics department prior to that.
"He would be missed, but Janet Yellen, if she's the Fed chair, has shown many leadership qualities and would likely carry the torch quite well. The market's confident that the bench is deep and what Bernanke's done has been institutionalized. The rules-based approach is institutionalized," Crescenzi said.
Rupkey said Yellen helped push through the balancing of the Fed's two mandates and the communication of them.
"Based on the interplay between the inflation goal and the full-employment goal, we can see for ourselves that the entire committee — or at least a few members — they are not going to raise rates until 2015," Rupkey said. "Maybe this is a good time to shift the chairman because I can't see a lot happening in Fed policy over the next couple of years."
Jeffrey Kleintop, chief market strategist at LPL Financial, said he expects to see Yellen giving the keynote in Jackson Hole this year.
"I think it's one of those things where Yellen will be the speaker. There will be a clear tip to the market that they're on top of this, and they're going to make the six months' transition smooth," he said. "And Bernanke's done when his term comes to an end. The market has so much faith in central banks right now. I think that could begin to shake confidence if they're not sure who is in charge."