Cohn is Trump's director of the National Economic Council, a post that gives him broad influence into many of the White House's domestic and foreign policy decisions. As chair of the Federal Reserve, one of the government's most powerful institutions, Cohn would help determine the course of the nation's economy by setting interest rates that influence the cost of borrowing and lending.

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Trump said he could also decide to reappoint Fed Chair Janet Yellen to a four-year term when her term expires in February. He said he hasn't made a decision. Trump told the Wall Street Journal that he respects Yellen and that he'd "like to see rates stay low. She’s historically been a low-interest-rate person," he said.

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“I’ve known Gary for a long time, but I’ve gained great respect for Gary working with him, so Gary certainly would be in the mix,” Mr. Trump also told the newspaper. He added that there were "two or three” other contenders in the mix.

Cohn and Treasury Secretary Steven Mnuchin are leading the process of recommending whether to reappoint Yellen or pick someone else, so Cohn could be in the position of deciding whether to recommend himself for the job.

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Cohn rose to the second-highest job at Goldman Sachs during a lengthy career and is considered an expert in financial markets. He has pushed for rolling back numerous regulations that were put in place after the financial crisis, and if he became Fed chairman he would have wide discretion to implement these changes.

Cohn was not part of Trump's team during the election but quickly joined after Trump won in November. He has emerged as one of the president's closest advisers, and his big, Wall Street personality meshes well with the president, advisers have said. Still, he has expressed frustration with Washington to friends in New York and his long-term plans in the administration have remained unclear.

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If Trump does decide to nominate Cohn, the Senate confirmation process could be bumpy. Cohn is considered a Democrat by many Republicans, but many Democrats have also expressed resistance to support anyone who has a Goldman Sachs background. Cohn's views on monetary policy are less well known than his views on banking regulation, and the Fed is in a period now of slowly raising interest rates.

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The Fed raised its benchmark interest rate by a quarter-point in June, the third such increase in six months. While investors widely expect the Fed to remain on hold at the conclusion of its two-day meeting this week, the central bank has forecast a steady pace of rate hikes for the next few years. It also plans to begin unwinding the huge balance sheet it accumulated after the financial crisis, a measure that could also push up the cost of borrowing.

Trump has said that, as a property developer with outstanding debt, he “always loved” low interest rates. During the campaign, he criticized Yellen for keeping rates low to benefit the Obama administration and called the Fed “very political” – charges Yellen has denied.

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But after his election, Trump appeared to warm to her. In an April interview with the Wall Street Journal, Trump said he liked and respected Yellen and did not rule out reappointing her.

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Trump has the opportunity to reshape the Fed’s leadership through appointments. The administration announced July 10 that it would appoint Randy Quarles, a former Treasury official and private equity investor, to one of three open seats on the Federal Reserve’s board of directors. Yellen’s term as Fed chair will expire on Feb. 3, 2018, while Fed vice chairman Stanley Fischer’s will end in June.

Past presidents have often sought continuity by keeping the Fed chairs of their predecessors. Former Fed chairs Paul Volcker, Alan Greenspan and Ben Bernanke were all reappointed to a subsequent term by a president of the opposite party.

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A change in leadership could be an unwelcome source of disruption to the markets, as the Fed continues with its plan to gradually lift interest rates to a more normal level in the next few years, said Greg McBride, chief financial analyst for Bankrate.com. “There’s a strong possibility we’re going to be changing pilots on final approach, which introduces another risk factor,” he said.

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But Brian Coulton, chief economist at Fitch Ratings, said he believed a change in personnel was unlikely to disrupt the Fed’s plan to continue hiking rates. “This desire for normalization is probably something that is quite deep seated in the institution. I don’t think that would be a very easy thing to change if someone new was to come in, even at the top.”