The Black Cats are the latest side to be linked with a move for the midfielder, who endured a disappointing debut season in English football last term.
Barnsley already have a battle on their hands to keep hold of striker Kieffer Moore, with three unnamed Championship clubs interested in the player.
The Reds signed Moore for £750,000 from Ipswich Town back in January and he is contracted at Oakwell until 2021.
Former Scottish youth international Mallan made just nine appearances after leaving St Mirren last summer.
Ross was Mallan’s manager in Scotland and the 42-year-old is a big admirer of the player - who kept his struggling Buddies side up with six goals in his final 10 games for the club.
New Sunderland boss Ross, who was interviewed for the Barnsley job before Jose Morais took over, went on to secure St Mirren promotion to the Scottish Premiership before taking on the rebuild mission at the Stadium of Light.
Ross faces stiff competition from Hibernian, with Neil Lennon keen on offering the player a route back to Scotland.
Barnsley will travel to Guiseley AFC for their annual pre-season friendly on Tuesday July 31.
New boss Daniel Stendel will take a mix of under-23s and first-team players to the game at Nethermoor.