The pick could be announced as soon as the coming week, the sources said.
The White House is waiting for his clearance information to be completed before announcing the choice, one of the sources said.
Cipollone met with White House officials earlier this week to discuss the counsel’s job, a different source familiar with the situation told CNN.
But sources caution nothing is final until an announcement is made.
Asked Saturday if Cipollone is going to be the next White House Counsel, Trump told reporters, “I haven’t named the new White House counsel. But over a short period of time, I will.”
When reporters followed up, asking if Cipollone was at the top of the list, Trump responded, “Pat’s a great guy, I don’t want to say, but he’s a great guy. He’s very talented and he’s a very good man, but I don’t want to say yet.”
Cipollone came to know Trump after advising the President and his legal team on the special counsel’s investigation, a source familiar with the matter previously told CNN.
A separate source said Cipollone has spent some time with the President in that role since the end of Supreme Court Justice Brett Kavanaugh’s confirmation hearings.
The tasks of the White House counsel’s office are expected to change dramatically if the Democrats take control of the House of Representatives and launch investigations and issue subpoenas.
McGahn is hoping to leave within the next couple of weeks, a source said. He has been expected to leave for months.
Another contender, Emmet Flood, will remain in place as a Russia-focused internal lawyer, one of the sources said.
A seasoned litigator, Cipollone is a partner at the Washington law firm Stein, Mitchell, Cipollone, Beato and Missner.
CNN’s Amanda Golden and Caroline Kelly contributed to this story.