Ms. Palin appeared to have reservations of her own. And the tension with Mr. Ailes, which was more nuanced than known publicly, would help open the door at the network for Mr. Trump.

She told people close to her at the time that Mr. Ailes made her uncomfortable, especially the way he commented on her looks. “He’s always telling me to eat more cheeseburgers,” she told one member of her staff.

Once, after a private meeting in Mr. Ailes’s office at the network’s headquarters in Midtown Manhattan in 2010, she came out looking white as a ghost.

Mr. Ailes’s assistant had asked that the aides and family members traveling with her wait outside so the two of them could meet alone. And when she emerged, according to the former staff member who was there, she said, “I’m never meeting with him alone again.”