The Clarets are on the search for a new first-team manager after parting company with Sean Dyche in the middle of April.
Burnley have persisted with caretaker manager Mike Jackson, overseeing five matches where he has won three, drawn one and lost one.
The Lancashire club sit above the drop zone on goal difference, ahead of Leeds United, but they do have a game in hand over the Whites.
Any new prospective manager would be expected to join at the end of the season, given there are only three games left in the campaign and the board’s faith in Jackson.
It will also be dependent on what division that Burnley find themselves in, as they could either be in English football’s top flight or the Championship.
Should they find themselves relegated then emerging as a key candidate for the role is League Two promotion winning boss Barton.
Barton had two spells at Burnley, playing in their 2015/16 Championship winning season, as well as coming back for the second part of their Premier League season in 2017.
The 39-year-old has quickly risen as an outside candidate at 40/1 to become the firm choice at 11/8.
It comes just days after the former Queens Park Rangers midfielder led the Gas to a dramatic promotion, overturning Northampton’s five goal deficit to get back to League One on the virtue of goals scored.
Barton was asked almost a month ago of his opinion on the sacking of Dyche, who became a close friend following their time together at Burnley.
He said: “From my experience there, I’m not in the dressing room, I’m not in the club so I don’t know what’s gone on, but I would think you’d have a much better chance with Sean Dyche, Ian Woan, Steve Stone and Tony Loughlan and Billy Mercer in the building than Burnley would without them.
"He’s been an incredible manager for Burnley as a town and a football club. So, hopefully, the next man who comes in can take it on to the next stage but it’s going to be a very tough act to follow that’s for sure."
Also listed amongst the favourites are former Manchester City defender Vincent Kompny, as well as Wayne Rooney, Chris Wilder, Bodo/Glimt boss Kjetil Knutsen as well as Jackson, who could take the job on a permanent basis.
As well as Barton being linked with a job, an old rivalry could be renewed as former Celtic midfielder Scott Brown has emerged as a key contender for the vacant Fleetwood Town role.
Barton’s old job is up for grabs after caretaker boss Stephen Crainey successfully guided them to safety after replacing Simon Grayson following his departure.
Fleetwood are now in talks to find a successor and Brown, who was the main rival of Barton during his time in the Old Firm Derbies, could be set for his first job in management, with an old rivalry potentially renewed.
The 36-year-old left his coaching role at Aberdeen and retired as a player, he also has talks over the vacancy with Scottish Championship side Raith Rovers.