The Clarets were not even listed as candidates at one stage, but are now 2/1 after leapfrogging Chelsea (6/1), Sporting Lisbon (7/1) and Lille (8/1).
Hart has spent the past two seasons away from Manchester City, on loan to Torino and then West Ham United.
And the 31-year-old continues to find himself behind Ederson Moraes and Claudio Bravo in Pep Guardiola's pecking order.
Burnley, meanwhile, are currently without Nick Pope and Tom Heaton, leaving third choice Andreas Lindegaard primed to start Thursday's Europa League qualifier against Aberdeen.
And it is plausible that a new goalkeeper will be sought by Sean Dyche ahead of next week's transfer deadline.
Already based in the north west, Burnley could be a convenient move for Hart - who earns £87,000 a week according to the MEN.
Could - and should - Burnley turn to the Manchester City goalkeeper?