The 32-year-old has been linked with a move away from Vicarage Road following Watford’s relegation to the Championship last season.
Turkish Super Lig outfit Fenerbahce are one club who are interested in acquiring the services of the centre-forward in the current transfer window.
The Yellow Canaries are said to be prepared to match Deeney’s estimated £100,000-a-week wage and offer him a three-year contract.
However, Fenerbahce are yet to make a formal offer for Deeney, who is claimed to be Watford’s highest-earning player.
It remains to be seen if the Turkish top flight club can strike a deal with the Hornets for the striker.
Deeney has also been linked with West Brom and Tottenham Hotspur, while Burnley boss Sean Dyche admires his talents.
The striker helped Watford to win promotion to the Premier League and the Hornets have to consider if he could do the same if they kept him.