Roger Ailes attends a special screening of "Kingsman: The Secret Service" in New York in 2015. Charles Sykes/AP/Press Association Images If the New York magazine is to be believed, Roger Ailes' Fox News career is hanging by a thread following the explosive sexual-harassment lawsuit filed against him by former anchor Gretchen Carlson.

The allegations in Carlson's lawsuit have been disputed by Ailes, who has called the accusations "false" and "offensive," and several Fox personalities have come out to express support for him, including Fox Business Network host Neil Cavuto and Fox News anchor Kimberly Guifoyle.

21st Century Fox said Monday that the matter was "not yet resolved." But it has not stopped widespread speculation about who might replace the Fox News chief executive in the future.

Here are the early frontrunners rumoured to be in line to take Ailes' job at a key part of Rupert Murdoch's global media empire.

1. Bill Shine, senior executive vice president, Fox News.

Bill Shine at book launch in 2015 Michael Loccisano/Getty Shine was cited by Variety, Forbes, Financial Times, and Bloomberg as Ailes' potential successor.

The FT said Shine would continue much of Ailes' work after being one of his closest allies during his 19-year tenure at Fox News. New York University professor Al Lieberman told the newspaper that Murdoch's view often is: if it's not broke, don't fix it.

But Variety speculated that Shine might be a better interim option and the "Murdoch family would likely look elsewhere for a stronger hand."

2. Jay Wallace, executive vice president, news and editorial, Fox News.

Wallace has been referenced as another strong internal contender.

He has worked at Fox News since its inception in 1996 and was promoted into his current role last year, which involved overseeing “The Real Story with Gretchen Carlson” before her contract expired last year.

3. John Moody, executive vice, executive editor, Fox News.

Another potential internal candidate, according to Variety, Moody has been in his current role for more than four years. He was the former Vatican correspondent and Rome bureau chief for Time magazine.

4. Jesse Angelo, chief executive and publisher, New York Post.

Jesse Angelo (left) with Rupert Murdoch last year. Natalie Ravitz/Tumblr Variety speculated that the Murdochs could look to another part of the News Corp empire to find a replacement for Ailes.

Angelo was one name it put forward after a long career in journalism in the Murdoch machine. He was the founder and editor in chief of The Daily, News Corp's failed tablet news app.

5. David Rhodes, president, CBS.

Rhodes extended his contract with CBS to 2019 only last year, but the Financial Times said he could be a potential replacement for Ailes.

He started his career at Fox News in 1996 and rose to become vice president of news. Deadline said he is known for his "his non-flashy, it’s-about-the-news approach."

6. Jeff Zucker, president, CNN Worldwide.

Along with Rhodes, Zucker is another high-profile external executive being linked with Fox News' top job, the FT writes.

He has been CNN's president since 2012, before which he was credited with making NBC's Today show a hit with audiences.

Zucker has been critical of Fox News in the past. In 2014, he said Ailes' biography confirms that "the Republican Party is being run out of News Corp headquarters, masquerading as a cable channel.”