Trump on Sunday called the story false, but The New York Times, which was the first to report it, said it was standing behind its account.
If an agreement was reached, Flood would not replace the White House's current lawyer Ty Cobb, according to The Times.
Cobb, however, has told friends that he views his position as temporary and does not expect to stay on the job much longer, the report said.
Flood is reportedly the only person the White House has been in contact with about the leading role.
Flood's law firm, Williams and Connolly, represents Hillary Clinton, including in the email scandal, one possible reason that sources told Reuters previous year the firm turned down representing Trump as outside counsel.
Kentucky edges No. 13 Tennessee for SEC championship
I don't need to tell Vol fans not to take this opportunity for granted. "We usually come with a lot of energy". If the Vols win their first round game, they would play the victor of Miami (FL) and Loyola Chicago.
Flood is a partner at the D.C. -based litigation firm Williams & Connolly LLP, which represents Hillary Clinton.
Flood also spent two years in the White House Counsel's office, where he handled executive-privilege related disputes for President George W. Bush's administration to congressional investigations and other inquiries.
Dowd told Reuters on Saturday that he did not know anything about Flood possibly joining the White House legal team. Mr. Flood declined to comment.
Neither the White House nor Flood returned a request for comment.
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". Marc Kasowitz, a NY litigator with a combative reputation who had represented Trump in business disputes, was replaced as head of Trump's personal legal team by John M. Dowd, who had deeper Washington experience.
Cobb is apparently nearing the end of fulfilling his main responsibilities - turning over documents and coordinating interviews between White House aides and prosecutors.
Trump then turned his focus directly on Haberman, making good on the promise of the ellipsis that concluded his (at this point nine-minute-old) tweet.
Turnbull Says Trump Promised Him Exemption on Metals Tariffs
The tariffs are created to counter cheap imports, especially from China, which Trump says are undermining US industries and jobs. The report found that the "quantities and circumstances of steel and aluminum imports threaten to impair the national security".
Nick Gordon Arrested For Domestic Violence
Brown told Rolling Stone that the wrongful death lawsuit wasn't enough and he wanted to see Gordon punished. There were also allegations of domestic violence in the house just before she was found dead in a bathtub.
In South Korea, Surprise and Skepticism at Talks
That's why many analysts, foreign policy experts and Pentagon officials are skeptical about the meeting, if it goes ahead. Some American officials are voicing concern that the talks will weaken the global pressure campaign against North Korea.
Messi Out Of Barcelona Squad For "Personal Reasons"
He then announced in February that a third son would be called Ciro, with Mateo and Thiago set to get a little brother. He wrote on Instagram: "Welcome Ciro!" Barcelona have a 10 point La Liga lead ahead of their meeting with Malaga .
Cent Clowns Juelz Santana, Twitter Got Jokes As Well
Once the gun was detected, Santana ran out of the airport and quickly hopped into a cab before fleeing the scene. According to sources, the Harlem rapper attempted to go through security at Terminal C with a gun.