Airey, who joined Sky from Five where she was chief executive, worked at Channel 4 as head of arts and entertainment before joining what was then Channel 5 in 1996.
Channel 4 chairman Luke Johnson said at the weekend that the net would be cast wide in search for a replacement for Thompson and that outside candidates would be considered.
The list of candidates in the running for the job alongside Airey include Kevin Lygo, Channel 4's director of television, and John Willis, the BBC's director of factual and learning who was an outsider in the race for the BBC job.
Johnson said that the appointment would be made as quickly as possible with only a "relatively short list of capable people" from within the media industry up to the job.
He said the right candidate needed a combination of experience in the media sector and a business track record.
"It could be someone from outside but they would have to be in tune with what Channel 4 stands for and its values. Channel 4 is unique -- part private sector, part public sector," he said.
Airey, 44, stunned the industry when she joined Sky in September 2002. She had been hotly tipped to replace David Liddiment and take up the chief executive's job at ITV as director of channels.
Her hiring was a major coup for then BSkyB chief executive Tony Ball and was seen as a clear sign that Sky was looking to secure its position as the UK's sixth channel.
While at Five, she reversed a reputation for cheap programming that relied on "football, films and fucking" and instead introduced quality US dramas, arts and historical programming, which has helped to consistently push Five's audience share.
Before leaving for Sky less than two years ago, Airey spent six-and-a-half years at Five having joined in 1996 as director of programmes.
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