Reports in The Scotsman's sister paper, the Sheffield Star, Dawson is attracting interest from Blackburn Rovers, Derby County and Preston North End as well as the Light Blues.
The 24-year-old, who is a graduate of the Owls' youth system, is contracted at Hillsborough until the end of the current campaign and it remains to be seen if he will be offered a new deal.
Dawson has made 44 appearances for Wednesday, keeping 15 clean sheets, and has also had spells on loan at Alfreton, Wycombe and Chesterfield. The former England Under-19 international is currently playing second fiddle to Keiren Westwood, while the club has two other goalkeepers in the first-team squad in Paul Jones and Joe Wildsmith.
Wednesday could be happy for Dawson to leave in January in order to get some sort of fee for the 6ft 4in shot-stopper.
Hladky claims Rangers interest but eyes move south
Meanwhile, St Mirren goalkeepr Vaclav Hladky has claimed Rangers are interested in signing him - but the former Slovan Liberec No.1 has his heart set on a move to the English Championship as a stepping stone to the Premier League.
In quotes published by the Daily Record, the Czech 'keeper said: “Rangers are interested in me and some negotiations are under way.
"But playing in England is the big dream. The biggest dream is to play in the Premier League but that is a big jump.
“I know I will need to make another step first on the way. I’m not crazy enough to think I can go from St Mirren directly to the English top division.
"Going to the Championship would be ideal for me. That is preferable to staying in Scotland.”
Comings and goings at Ibrox?
Rangers have been linked with several goalkeepers since the summer, with reported interest in Dawson and Hladky following on from last week's link with Preston goalie Declan Rudd.
During the summer QPR's Seny Dieng was also touted for a move to Ibrox after featuring on loan for Dundee last season while the Gers were also credited with an interest in Jonathan Klinsmann, the goalkeeper son of former Germany international Jurgen.
With veteran Allan McGregor the first-choice at Ibrox and in line for a new deal, Rangers' supposed interest in a new goalkeeper could suggest Wes Foderingham is on his way out.
The former Gers No.1 has been linked with a move to Major League Soccer amid claims he could move on in search of first-team football while Nottingham Forest failed with a bid for the former Swindon goalie during the summer transfer window. DC United are understood to be keen on the 28-year-old - could Rangers' increased interest in goalkeepers mean the end for Foderingham?