The White House claimed the story was incorrect.
"Obviously he'd be very, very good, but let's wait and see because H.R. McMcaster's still in the job and he's doing a really fine job", Rice, who also served under Bush, added. Trump took to Twitter to push back on McMaster's remarks at the Munich Security Summit last month.
However, White House spokeswoman Sarah Huckabee Sanders said on Wednesday Hicks' departure was not related to her testimony.
Trump publicly took a rhetorical swing at McMaster last month, when he asserted that McMaster "forgot to say" during one public appearance that the 2016 U.S. presidential election was "not impacted or changed" by Russia's meddling.
Raj Shah, the principal deputy press secretary, said that the White House frequently faces "rumor and innuendo about senior administration officials". The president has also begun considering candidates to succeed McMaster as national security adviser, a post which does not require Senate confirmation, the sources told ABC News. There are no personnel announcements at this time'.
The two have never gotten along, and Trump continues to chafe at McMaster's demeanor when he briefs him, feeling that he is gruff and condescending, according to a source who is familiar with his thinking. The White House did not respond to requests for comment. Trump's first national security adviser, Michael Flynn, resigned amid controversy over his contact with Russian officials within a month of taking the job.
Mr McMaster and Mr Trump butted heads recently, which could prove to be one of the deciding factors in his dismissal.
McMaster became national security adviser in February 2017 after the resignation of Michael Flynn.