It comes less than two weeks after former presenter Gretchen Carlson sued Mr Ailes for sexual harassment and wrongful termination, claims he denies.
The New York Times, citing a source, said Mr Ailes and parent company 21st Century Fox were in the advanced stages of talks on his departure.
The company would only say Mr Ailes was at work and a review was ongoing.
A report in New York magazine, citing anonymous sources, says lawyers for 21st Century Fox have given Mr Ailes a deadline of 1 August to resign or face being fired.
The lawsuit filed by Ms Carlson claims she was fired after refusing his sexual advances.
She worked for the conservative-leaning US network for 11 years before her contract expired in June.
Mr Ailes strongly denies the allegations and says Ms Carlson filed the suit in retaliation for not having her contract renewed.
"This defamatory lawsuit is not only offensive, it is wholly without merit and will be defended vigorously," he said.
Further allegations have since surfaced in US media that Mr Ailes sexually harassed another Fox News presenter, Megyn Kelly.
According to New York magazine, Ms Kelly has told investigators that Mr Ailes made unwanted advances towards her about 10 years ago when she was a correspondent.
A lawyer for Mr Ailes, Susan Estrich, denied those claims in a statement on Tuesday.
Fox News Channel boss Roger Ailes is negotiating his departure from the network, according to US media reports.