HITC Sport understands that that the Clarets have made contact regarding a deal for the Republic of Ireland international.
The 27-year-old has been told by Bhoys boss Ronny Deila that he does not figure in his plans for the first-team and is free to move on – as revealed by HITC Sport in July.
A host of clubs in England have been alerted to the availability of the Republic of Ireland international, who now looks set for a return south of the border.
But now HITC Sport understands that Sean Dyche's Burnley are leading the chase and that they have made contact with the Scottish Premiership champions about doing a deal for Stokes.
Stokes, who has scored 79 goals in 191 games for Celtic, has one year left on his current contract.