Reuters reported Monday, citing sources familiar with the matter.
Cobb met with the president about a week ago, another person told the news wire.
Cobb, who formerly served as a federal prosecutor, currently works as a defense lawyer at the law firm Hogan Lovells.
{mosads}White House spokeswoman Lindsay Walters told Reuters that the Trump administration has “no announcement at this time.”
Cobb did not respond to the news wire’s request for comment on the matter, except to say he is on vacation.
If hired, Cobb would reportedly serve in a top-level position within the office of White House counsel Don McGahn.
The administration has been looking to fill the position, which would deal with the congressional and federal Russia probes, one of the people familiar with the matter told Reuters.
The investigations are looking to see whether the Trump campaign colluded with Russia as well as whether the Kremlin interfered in the 2016 presidential election.
Trump has repeatedly denied that his team colluded with Moscow.
Trump also has a separate legal team to represent him amid the probes, led by his longtime New York-based lawyer Marc Kasowitz.
Other lawyers such as William Burck and Emmet Flood have declined to accept the role that Cobb is now being considered for, people familiar with the matter told Reuters.
The news wire reports that Cobb is a distant relative to Ty Cobb, a famous baseball player who shares the same name.
Cobb earned his law degree at Georgetown University after graduating from Harvard University.