The 36-year-old Kompany left his head coach role at Belgian giants Anderlecht on Wednesday, which had bookies scrambling to slash his odds to 1/20 for the Turf Moor hotseat.
Kompany guided his boyhood club Anderlecht, who he has managed since 2019, to third in the Belgian league this season, but it now appears the lure of recently relegated Burnley could be too hard to resist. He spent 11 years in the region after joining City, and wife Carla is from Manchester and their three children were all born in the city.
Anderlecht confirmed his departure after three years and in a statement on the club website, Kompany said: "Today only one feeling prevails: I am proud that I was able to start this new chapter at the club of my heart. I have now been a player and a coach of RSC Anderlecht, but above all I remain a loyal fan."
Joey Barton, Wayne Rooney and Chris Wilder are next in line on the bookies list to take over at Turf Moor, with Mike Jackson, who was caretaker manager after Sean Dyche was sacked by the club in April, fifth favourite.
If Kompany was appointed he faces a stiff task to return the Clarets to the Premier League at the first time of asking, especially as the club have to repay a "significant proportion" of a £65million loan taken out during ALK Capital's leveraged takeover in December 2020.
Burnley also have nine first-team players out of contract, while key players England goalkeeper Nick Pope, Dwight McNeil, Maxwel Cornet and Wout Weghorst will all be heavily linked with moves in the summer.