Speaking to reporters on Friday, Trump said he would make his decision in the next two to three weeks.
The Wall Street Journal reported that Trump had also met with Jerome Powell, the only Republican on the Fed board. But, that he is also looking at other candidates including Gary Cohn, the head of his National Economic Council.
President Trump has said that he is considering re-nominating Chairwoman Janet Yellen when her term ends in February.
Mr Warsh is married to Jane Lauder, daughter of Mr Trump's friend Mr Ronald Lauder and a global brand president at the cosmetic company founded by her grandmother, Estee Lauder.
Mr Warsh, 47, who served as Fed governor from 2006 until 2011, didn't respond to a request for comment through the Hoover Institution, where he is now a fellow. Her father, Ronald Lauder is a longtime Trump friend.
Real Madrid hit with another major injury blow ahead of Espanyol match
But, as always, when a player has even a small problem then we don't want to risk it. See: Real Madrid v Espanyol live stream .
Treasury Secretary Steve Mnuchin said this month that Yellen was one of "a lot of good people" under consideration.
Yellen has said she would consider accepting a second term as the nation's top banking official, though it's unclear whether Trump would be willing to keep her on.
"In terms of policy, I think it is fair to say that Warsh is less dovish than Yellen", said Stephen Stanley, chief economist at Amherst Pierpont Securities in NY.
It is possible that Trump's four meetings were with Yellen, Cohn, Warsh and Powell, though Trump did not provide names.
According to a New York Times report last weekend citing administration officials, the White House had created a short list of candidates for the post. The US president may have to fill more than one spot on the bank. His nomination is pending before the Senate.