Bill O’Reilly left Fox News this week for a long-planned vacation to Italy and the Vatican, with his fate in the hands of a Murdoch family calculating the risks and rewards of keeping him on or forcing him out of the network.

Facing a boycott by advertisers, unrest inside the company, protests outside Fox News headquarters and public calls for human rights investigations into company culture, Rupert Murdoch and his sons, Lachlan and James, are reckoning with the fallout of a sexual harassment scandal that has once again engulfed Fox News and its parent company, 21st Century Fox.

The Murdochs are awaiting the results of an investigation into Mr. O’Reilly’s conduct before making a decision about whether he will stay or go, two people briefed on the plan said Wednesday. The law firm Paul, Weiss, Rifkind, Wharton & Garrison has been enlisted by 21st Century Fox to investigate, and although the probe started with a review of accusations made by Wendy Walsh, a former guest on Mr. O’Reilly’s show, the law firm is expected to expand its purview if other issues arise, the people said.

The calculation on whether to keep Mr. O’Reilly is complex and rests in part on a generational divide between Rupert Murdoch, 86, who controls 21st Century Fox, and Lachlan, 45, and James, 44, who have top leadership positions at the company.