It is thought that she has been offered a job overseeing a group of US-based Discovery channels.
If she goes it will be a coup for Discovery, which has been trying to poach Root for several months. She is understood to have visited the broadcaster's Washington headquarters a number of times.
According to reports, a BBC source said that Root was certainly interested but that she had not said yes. The source said: "With Greg gone everything is now up in the air."
The BBC has declined to comment on the reports that it might lose its much-rated BBC Two chief.
The BBC and Discovery have a co-production deal, which means that Root has had numerous dealings with the company. Last year this was extended to give the Discovery Channel access to BBC news.
Discovery first signed an agreement with BBC Worldwide in 1999 for access to moving images, mostly from the BBC's archive of natural history programming. The two companies also have a joint venture to produce programming that is broadcast by both companies.
Root is a well-respected figure in the industry. She joined the channel in 1999 and since then has been voted as one of the most influential women in the UK in Good Housekeeping magazine.
Under her reign, BBC Two was awarded channel of the year at the Guardian International Media Festival last year. She was previously head of the Independent Commissioning Group.
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