Some Republican lawmakers, particularly in the House, are quietly lobbying the White House to appoint Taylor, a longtime hero in GOP circles, according to three people familiar with calls and discussions who were not authorized to speak publicly. Vice President Pence also likes Taylor, according to several White House officials not authorized to speak publicly. The Wall Street Journal editorial page, influential among conservatives, endorsed Taylor or Warsh on Friday.

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But Trump is also contemplating keeping Yellen, with whom he met this week, as Fed chair. Unemployment has fallen to a 16-year low during her tenure. Top economic adviser Gary Cohn and current Fed governor Jerome Powell, a Republican, are also on the president's shortlist for the job.

Many on Wall Street argue what's needed most right now at the Fed is continuity and low rates. Treasury Secretary Steven Mnuchin, a former Goldman Sachs executive, is pushing Powell as the Goldilocks option that would allow Trump to keep the Fed on a steady and familiar course without reappointing Yellen, a Democrat, according to several people close to Mnuchin who spoke on the condition of anonymity because they were not authorized to discuss the internal deliberation.

Trump said Friday that “it is in my thinking” to appoint both Powell and Taylor to the Fed, although he didn't say which one he would pick for the top post. “They're both very talented people,” he told Fox Business in an interview.

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The White House, which has confirmed the list of five finalists, says Trump aims to announce a nominee before the president leaves for Asia on Nov. 3.

The decision could have wide ramifications for the economy — and politics. Changes in Fed policy can affect economic growth and markets, sometimes in unpredictable ways. The selection of a new Fed chair also comes in the midst of heated debates on Capitol Hill over a massive overhaul of the tax code. Alan Greenspan famously broke with the Fed's tradition of staying out of politics to endorse George W. Bush's tax cuts in 2001, lending his imprimatur to the successful effort.

Yellen’s tenure, like that of her predecessor, Ben Bernanke, has been marked by unprecedented efforts to fight unemployment and stimulate economic growth in the aftermath of the 2008-2009 recession. Yellen continued a trend of holding interest rates at historic lows for nearly a decade and even employed never-before-used monetary maneuvers in an effort to accelerate the recovery.

Warsh and Taylor argue that Yellen has gone too far in pushing for economic stimulus, saying she has left the United States vulnerable to the type of out-of-control inflation it faced in the 1970s and early 1980s. If Trump tapped either for the job, the new chair would probably push the Fed to raise interest rates much more quickly to limit the risk of inflation, a move that could startle markets and potentially slow the economy.

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Appointing one of them would be a long break with tradition, as sitting Fed chairs have been reappointed by presidents of opposing parties ever since Ronald Reagan did so with Paul Volcker, a Jimmy Carter appointee.

“She’s done a magnificent job,” says Allen Sinai, president of Decision Economics and a longtime adviser to both political parties. “The main problem is she is not a loyal Republican. Janet is a liberal Democrat. She doesn’t hide this.”

Many in the GOP think the Fed is playing too big of a role in the economy, especially after Yellen and Bernanke took the Fed’s assets from $900 billion to $4.5 trillion today.

“I would want someone who is explicit about ditching the framework they are operating under now,” says Norbert Michel, director of the Center for Data Analysis at the conservative Heritage Foundation.

The Wall Street Journal editorial board's endorsement of Taylor and Warsh on Friday stated they would be “change agents” at the Fed. The Journal argued that Taylor and Warsh, both Republicans, would be a lot more supportive of Trump's tax cuts than Yellen or even Powell, the token Republican President Barack Obama appointed to the Fed. But some economists find that argument puzzling, since Taylor and Warsh could probably raise interest rates quickly, a move that would tap the brakes on growth.

“It’s not in the Trump administration’s best interests to have a series of rapid rate hikes,” says Seth Carpenter, chief U.S. economist at UBS, an investment bank.

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Presidents typically consider multiple candidates when selecting a Federal Reserve chairman, but the candidates usually have similar visions for the bank. When Obama selected Yellen, the other major contender was Lawrence H. Summers, a former treasury secretary and top White House economic adviser who shared many policy views with Yellen despite differing in style.

Trump's own views on America's central bank have swung wildly in the past year and a half. On the campaign trail, he slammed Yellen for keeping rates artificially low “because she’s obviously political and doing what Obama wants her to do.” Now that he is in the White House, Trump says he “like[s] a low-interest-rate policy” and “respects” Yellen.

What complicates the decision for Trump is that he wants the next Fed chair to do more than craft monetary policy; he wants that person to lead the charge on rolling back regulations on banks, including the Dodd-Frank Wall Street Reform Act put in place after the crisis. Yellen has repeatedly stated that she thinks banks are much safer today because of regulations, although she has expressed some openness to making tweaks to Dodd-Frank, especially for smaller bankers.

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Trump has an unusually large opportunity to shape the Fed. In addition to picking the chair, he gets to fill four openings out of seven Fed governor seats, meaning if he wanted to go in a radically different direction, he could load the board with people who share those views. If he doesn’t reappoint Yellen and she steps downs, Trump would then get to make five appointments, an unprecedented number in a short period. He has already filled one seat with Randal K. Quarles.

Powell has emerged as a strong candidate, meeting many of the qualities that would seem to align with Trump's preferences. PredictIt, a Wall Street betting pool, has him in the lead. He’s been a Fed governor since 2012, helping steer unemployment to a 16-year low and the stock market to record highs. Powell would probably keep in place many of Yellen’s monetary policies, but he is a Republican and former partner of private equity firm the Carlyle Group who shares Trump’s views on pulling back regulations. Cohn, a former Goldman Sachs executive, is also seen as someone who would provide continuity, but with a lighter touch on regulation.

“There’s a pretty compelling argument for putting money on Powell getting the nomination,” says Michael Strain, director of economic policy studies at the right-leaning American Enterprise Institute.

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The decision will rank among the most important of Trump’s presidency, as the Fed chair wields tremendous power over the domestic and global economy. When a recession hits or the markets slide, it’s often the Fed that steps in first to intervene. The Fed operates independently of the White House, but the president gets to nominate Fed governors, including the chair, when there are openings. The Senate approves the president’s nominees.

“It’s the most important economic policy position in the world,” says Strain. “The importance is hard to overstate.”

Trump has recently fixated on the stock market, tweeting and speaking frequently about how it is at record highs. The Dow topped 23,000 for the first time ever this week. A Fed chair who aggressively hiked interest rates would complicate his goal to have a stronger economy and thriving market by the time he is up for reelection in 2020.

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“This is a president who loves low interest rates and happy markets, so this decision should be easy: Yellen, a no-brainer,” says Greg Valliere of Horizon Investments. “But he also hates regulations and must realize that Dodd-Frank reforms are going nowhere in the Senate, which means the Fed will have to lead the fight to kill regulations.”

The Fed and the White House have had tense relationships over the years. Former president George H.W. Bush says his Fed chair, Greenspan, caused his reelection loss by keeping interest rates high for too long, helping trigger a recession in 1990-1991. And former president Jimmy Carter also watched as then-Fed Chair Volcker worked ardently to stomp out inflation in the late 1970s, a move that helped shoot unemployment to almost 8 percent just as Carter was making an unsuccessful attempt at reelection.

Trump’s decision also comes at a critical time for the Fed. Yellen’s term ends in February, in the middle of the central bank’s efforts to wind down its economic stimulus program, return interest rates to more-normal levels after nearly a decade of historic lows and shrink its role in the economy. Completing that transition would give the Fed more tools to fight a future recession, but the next chair will be tasked with doing so in a way that doesn’t shock markets or undermine current growth.

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The five shortlisted candidates have each met with the president. While cameras aren’t allowed in the room, the White House has let the contenders be known, and Trump has made playful comments about Cohn’s status as a contender and how he “likes them all.”

“This feels a lot like ‘The Apprentice,’ ” says Rajeev Dhawan, director of the Economic Forecasting Center at Georgia State University. He says the process was mostly done behind closed doors in the old days, although he thinks Trump is wise to “take away the secrecy” on an appointment that is arguably on par with a Supreme Court nomination.

Cohn was seen as the ideal Fed chair choice for Trump for much of the summer, but he angered Trump when he criticized the president's remarks after the white supremacist rally in Charlottesville.

Warsh is seen as a GOP insider with ties to top Republican donors, but he’s younger than many of the other candidates and doesn’t have as much experience with Wall Street or academia. He’s a lawyer by training, although he did serve as one of the Fed governors from 2006 to 2011 and in the Bush administration.

Taylor, who worked in the George H.W. Bush administration, thinks QE, or quantitative easing, was a mistake. He says the Fed should act almost like a computer, raising or lowering interest rates when clear, transparent metrics are hit in the job market and inflation. His framework for setting interest rates has been dubbed the “Taylor Rule.” If it were in place now, interest rates would probably be higher than the current range of 1 to 1.25 percent.

But others argue that Taylor and Warsh wouldn’t disrupt the Fed’s slow and steady path of raising interest rates and unwinding QE. Once people get on the Fed board, they have a tendency to soften their views and not operate independently of the White House. All of the Fed’s key decisions are made by a committee of 12 people.

“You’re the Fed chair, but you’re not the dictator,” says Michel, of the Heritage Foundation.

Past presidents have often surprised the world with their Fed chair picks. In 2005, President George W. Bush was considering several prominent Republican economists for the job but ended up going with Ben Bernanke, an academic relatively unknown in Washington.

“Ben Bernanke was a surprise to me,” says Sinai, of Decision Economics, who went to the Bush White House several times for meetings about tax policy. “But history tells us, it turns out to have been one of Bush's best moves.”