The shot-stopper, a lifelong Wednesdayite, is out of contract in the summer.
But the Owls are keen to tie Dawson to fresh terms and have held discussions with his representatives.
Speaking at the beginning of December, Wednesday manager Garry Monk said: "I have spoken with the club and obviously he’s a player, with the age he is and where he’s at with the club, he’s been fantastic.
"Keiren [Westwood] has been the mainstay for this club and still has a hugely important role to play, but you have to think about the future as well, the long-term future.
"We have got Joey [Wildsmith] coming back to fitness so two really good young keepers. They are the future of the club. You want to keep them here.
“I have made it clear to them that they’re going to be a big part of what I’m doing here."
Dawson, who made his Wednesday debut in October 2016, has blossomed since taking over the No 1 jersey from Westwood last November. The academy graduate has started the last 11 matches on the bounce, racking up four clean sheets to help the Owls climb up to sixth in the Championship rankings.
His impressive form has caught the eye of a number of clubs, including Rangers, Derby County, Blackburn Rovers and Preston North End.