Derby County, Blackburn Rovers and Preston North End are believed to be closely monitoring Dawson's situation ahead of the transfer window reopening in January.
The 24-year-old, who has come through the ranks at Hillsborough, is also thought to be on the radar of Scottish giants Rangers.
Dawson's present contract runs out at the end of the season and it remains to be seen whether he will be offered a new deal.
Three years have passed since Dawson made his Wednesday debut as a half-time substitute against Brighton and Hove Albion.
Dawson, an avid Owls fan, was eventually handed the No 1 jersey in the first half of the 2018/19 season after then boss Jos Luhukay ostracised Keiren Westwood from the squad. He started 22 matches on the bounce, recording his first clean sheet in the Sheffield derby.
But Dawson lost his place in the side following the sacking of Luhukay last December and has largely played second fiddle to Westwood ever since.
Suspension and injuries to Westwood has given Dawson game time this season. Dawson has recorded three shut outs from his five starts and one substitute appearance, with boss Garry Monk admitting it was his "toughest" selection decision since taking over the reins to play Westwood over him in the Yorkshire derby with Leeds United last month.
Dawson is one of several players in the final year of his contract, including Sam Hutchinson, Steven Fletcher, Atdhe Nuhiu and Fernando Forestieri.
Monk said: “When I next sit with the owner, we will have those discussions and discussions about other things.
“Players find themselves in all different sorts of situations in my experience.
“If it is a case of where some of them are out of contract, it is their job to perform and earn a new contract. They have to do it with a real strong mentality and show us what they have got.
“But, of course, you would like to sort those situations out as quickly as possible but that will be done with the club’s blessing.
“Right now, I am not focused on that.”