Thompson, 66, is also a former chief executive of the New York Times and is a well respected name in US media circles.
According to the Semafor news platform the ex-BBC director general is a favourite to succeed Chris Licht, who was fired in June after a tumultous 14 months at the helm.
Since then Warner Bros. Discovery owned CNN has been led by a triumvirate of veteran network leaders on an interim basis.
Thompson joined the BBC as a production trainee in 1979 and quickly rose to edit programmes such as Panorama and the Nine O’Clock News. He became the 14th BBC DG in May 2004, a position he held until 2012 when he joined the New York Times. He led the newspaper until 2020.
He has also been CEO of Channel 4 and currently serves as chairman of the board of Ancestry, the Utah-based genealogy company.
A CNN insider told the New York Post that Thompson’s track record of transforming the New York Times from a print-centric business to a digital business is “exactly what this place needs.”
“It’s not a programming job, we need someone to come in and rethink how we approach things,”
In 2021 Thompson said in an interview “News feels like a particularly old-fashioned style of broadcasting aimed entirely at older audiences.
“I live in the US and [TV news] seems completely unchanged since the 1980s. I think it is in dead trouble.”
Thomson who is said to spending time at his home in Maine, was unavailable for comment. There was no comment from Warner Bros. Discovery.