By Lisa Richwine and Jessica Toonkel

(Reuters) - Rupert Murdoch, executive co-chairman of 21st Century Fox Inc <FOXA.O>, and his sons James and Lachlan agree that Fox News Channel boss Roger Ailes should leave the company but they have not settled on the timing, New York magazine reported on Monday, citing anonymous sources.

Former Fox News anchor Gretchen Carlson has sued Ailes, claiming sexual harassment. Ailes has denied the charges. Fox hired the law firm of Paul, Weiss, Rifkind, Wharton & Garrison to conduct an internal investigation.

"This matter is not yet resolved and the review is not concluded," 21st Century Fox said in a statement after the report surfaced. Julie Henderson, a spokeswoman for 21st Century Fox, declined to comment beyond the statement.

An attorney for Ailes did not immediately respond to a request for comment. A lawyer for Carlson declined to comment on the report.

New York magazine reported that two sources briefed on the investigation said that all three Murdochs "have settled on removing" the 76-year-old Ailes. Lachlan Murdoch is executive co-chairman of 21st Century Fox and James Murdoch is chief executive officer.

After reviewing the initial findings of the Paul, Weiss, Rifkind, Wharton & Garrison investigation, James Murdoch "is said to be arguing that Ailes should be presented with a choice this week to resign or face being fired," the magazine reported.

Lachlan "is more aligned with their father, who thinks that no action should be taken until after the GOP convention this week," the magazine reported.

Ailes was a consultant for several U.S. Republican presidents, including George H.W. Bush. He has been a confidant of 21st Century Fox Executive Co-Chairman Rupert Murdoch, who named him founding chief executive of Fox News Channel in 1996. Under Ailes' leadership, Fox News has become the top-rated U.S. cable news channel.

Ailes said in a statement after Carlson's lawsuit was filed that he would defend himself against "false" and "offensive" allegations.

Removing Ailes now, in the thick of a presidential election campaign, could pose a challenge for the network, said Rick Edmonds, a media business analyst at the Poynter Institute.

"The concept and the execution of Fox News is so closely associated with Ailes and his instincts and it is not obvious that someone can step in to do that," Edmonds said. "It will constitute a major challenge for them."

(Reporting by Lisa Richwine in Los Angeles and Jessica Toonkel in New York; Editing by Jeffrey Benkoe and Andrew Hay)