Flood's corporate profile outlines his experience in several high-profile legal fights, including the Clinton impeachment process and former Vice President Dick Cheney in a civil suit from former Central Intelligence Agency officer Valerie Plame.
Flood, who advised former President Bill Clinton in impeachment proceedings, met with Trump in the White House this past week, the person said, but no decision about the job has been made.
Flood, who declined to represent Trump last summer, would not replace Cobb if he took the job, according to the Times. Cobb has also reportedly told friends he does not expect to remain in his job much longer, and some advisers have suggested Trump invite his longtime NY lawyer, Marc E. Kasowitz, to take the lead again.
He tweeted out saying, "The Failing New York Times purposely wrote a false story stating that I am unhappy with my legal team on the Russian Federation case and am going to add another lawyer to help out".
Trump's personal lawyers, John Dowd and Jay Sekulow, have been negotiating with Mueller over the terms of a presidential interview, the newspaper reported.
The White House did not respond to a request for comment, and a spokesperson for the law firm could not be reached for comment.
If Flood joins the Trump team, then he would be changing his mind about representing Trump. He also represented former Vice President Dick Cheney in a civil suit brought by former Central Intelligence Agency officer Valerie Plame.
Trump claimed the report's author, Maggie Haberman, knows "nothing" about him. Flood was one of a half-dozen top litigators Trump reportedly tried to hire to replace Kasowitz previous year, but they all turned down the job.
Cobb is apparently nearing the end of fulfilling his main responsibilities - turning over documents and coordinating interviews between White House aides and prosecutors.
Flood's law firm, Williams and Connolly, represents Hillary Clinton, including in the email scandal, one possible reason that sources told Reuters a year ago the firm turned down representing Trump as outside counsel.
The Times reported that Trump is considering adding the lawyer that helped Bill ClintonWilliam (Bill) Jefferson ClintonShould the Rob Porter outcome set the standard?