The Black Cats parted company with Paolo Di Canio on Sunday after a poor start to the season left them bottom of the Premier League with one point from five games.
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"Of course I am flattered that I have been considered but I am really happy over here and like what I am doing," he told Sky Sports News.
"I think my growth as a manager has been massive and so I want to fulfil this task."
Zola started his managerial career at West Ham in 2008, but saw his contract terminated by the London club in May 2010 after narrowly avoiding relegation.
He subsequently took the reins at Vicarage Road in July 2012, and immediately guided Watford to the play-off final - where they lost 1-0 to Crystal Palace at Wembley.
Watford are currently fifth in the Championship, having lost just once in the league this season.