Sunderland parted company with Simon Grayson over a fortnight ago, but they still have not found a replacement.
Heckingbottom is said to be the Black Cats first choice though, and the 40-year-old would be interested in moving to the Stadium of Light.
Heckingbottom used to play for Sunderland, but he previously batted away suggestions that he was about to leave Barnsley.
However, Martin Bain has now been encouraged by Heckingbottom’s apparent interest, and Sunderland are expected to make a move soon.
Sunderland are prepared to pay compensation for their next manager, although they don't have a huge budget to spend.
Barnsley are currently eight places ahead of Sunderland in the Championship table, but the Black Cats do arguably have more potential as a club, if Heckingbottom can get them back to winning ways.
Sunderland currently have Robbie Stockdale in caretaker charge, with their next match coming against Millwall at the weekend.