Politico said sources in the White House, the Treasury Department and on Capitol Hill said that if Cohn decides he wants the job, he is likely to get it.
"It's Gary's if he wants it, and I think he wants it," Politico quoted one Republican whom it said was close to the selection process as saying.
A few Senate Republicans may express reservations about Cohn but he would probably receive widespread support, a senior congressional Republican aide said, according to Politico.
In response to a query from Reuters, White House spokeswoman Natalie Strom said: “Gary is focused on his responsibilities at the NEC.”
Cohn, a Democrat and former Goldman Sachs president, did not work on Trump's campaign and only got to know him after the November election.
Yellen took over from Ben Bernanke as Fed chair in February 2014 with the U.S. economic recovery from the 2008 financial crisis still on shaky ground. She has made no secret she puts a priority on growth in jobs and wages and a broad recovery in U.S. household wealth.
Yellen begins two days of congressional testimony on Wednesday on the Fed's semi-annual report on the state of the U.S. economy. Under her leadership the Fed has pared the massive stimulus it put in place to counteract the 2007-2009 financial crisis and begun raising interest rates. Later this year Yellen expects to begin trimming the Fed's $4.5 trillion balance sheet in a gradual and predictable manner.
"I do think the Fed's trying to set a course toward normalization that won't be knocked out by a new Fed chair," said Eric Stein, co-director of global income at Eaton Vance. Still, Cohn would be a very different kind of leader than Yellen or her immediate predecessor, Ben Bernanke, both of whom hold doctorates in economics.
Cohn began his career at Goldman as a commodities trader in 1990.
"He has more knowledge of financial markets than almost any Fed chair would have and less of a monetary policy background," Stein said.
During last year's election campaign, Trump accused Yellen of accepting orders from then-President Barack Obama to keep interest rates low for political reasons, and he said he would replace her as Fed chair because she is not a Republican Party member.
But in an interview with The Wall Street Journal in April, Trump said Yellen was "not toast" and that he respected her.
A Fed spokeswoman had no comment on the Politico story.
Yellen has said she plans to serve out her full four-year term as chair, which runs through Feb. 3.
Cohn would join several other Goldman alumni around the Fed policymaking table, including New York Fed President William Dudley, Minneapolis Fed President Neel Kashkari and Dallas Fed President Robert Kaplan.
(Reporting by Eric Beech; Editing by David Alexander and Leslie Adler)