According to a report in the Financial Times, Fincham, who is currently chief executive of production company TalkbackThames, has held informal talks with Channel 4 and the headhunter who is looking for a replacement for Thompson, who left earlier this month to take up the director-general role at the BBC.
Fincham is now being seen as the new frontrunner after Airey, managing director of Sky Networks at BSkyB, ruled herself out of the race following an informal meeting with the Channel 4 chairman Luke Johnson.
Her withdrawal from the race is a blow to Channel 4, which saw the former Five chief executive as the best-qualified candidate for the job.
Channel 4 is looking for someone who is familiar with the network's output and audiences, and Fincham has production credits on many of the broadcaster's best-known programmes. With an eye for edgy productions, he has helped bring shows such as 'Jam' and 'Brass Eye' to the screen, as well as 'The Armando Ianucci Show'.
However, it is not all about controversy for Fincham, who also has credits on the Stephen Poliakoff dramas 'Shooting the Past' and 'The Lost Prince'.
Fincham joined Talkback in 1985 when it was a company producing radio shows, ads and corporate videos. As managing director, he led the company's shift into television in 1989.
The BBC has agreed to pay the costs of the recruitment process after Channel 4 agreed to release Thompson early from his contract, which stipulated he work a six-month notice period.
Channel 4's director of television Kevin Lygo and Endemol chairman Peter Bazalgette have also ruled themselves out of the running.
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