Cipollone has already started the process of filling out necessary paperwork to replace outgoing counsel Donald McGahn, according to the Axios news website.
"He's not an obvious choice. Not a political guy like Don was. Not tied to the Hill and doesn't know all the senators like Don does," the source was quoted as saying by Axios.
In August, Trump announced that McGahn would leave the White House this fall after the confirmation of Supreme Court Nominee Brett Kavanaugh. Trump later stressed that his decision on Donald McGahn was not affected by the ongoing ‘Russiagate’ investigation by US Special Counsel Robert Mueller.
The US-based Media has reported earlier that some Trump allies were worried McGahn gave too much information to Special Counsel Robert Mueller during his interviews. Donald McGahn has served as general counsel for Trump's campaign and transition team.
Muller is investigating allegations that Russia interfered in the 2016 US presidential election. Russia has repeatedly denied meddling in the vote, calling such accusations "absurd."