Eyre, who is currently chairman of the Internet Advertising Bureau and was once the chief executive of ITV, is thought to be one of four or five candidates for the job. Others are believed to include Lord Alli, the chairman of rights holder Chorion, which owns Mr Men, Noddy and Miss Marple.
The channel is keen to have at least one woman on the shortlist and Dianne Thompson of Camelot has been linked with the post.
The Telegraph reports that a decision will be made within a fortnight.
Eyre was media director of Bartle Bogle Hegarty (BBH) in 1986. In the 1990s, he moved into commercial radio as chief executive of Capital Radio, and in 1997 became Chief Executive of ITV and later Pearson Television.
He worked with Fru Hazlitt at GCap until last year and it is rumoured that the pair could be reunited at Channel 4. The broadcaster has also begun looking for a new chief executive with digital media experience, who can help secure the channel’s financial future.
Hazlitt is one of the names in the frame, along with Kevin Lygo, C4’s director of television and content, ITV’s director of television Peter Fincham, UKTV chief executive David Abraham, and Jane Root, the former BBC2 controller.