• Chelsea sporting directors Laurence Stewart and Paul Winstanley are leading the recruitment process.
• Kompany's impressive season has also led to him being linked with the vacant Tottenham job.
Image: /BBC
Vincent Kompany is one of the managers on Chelsea's shortlist for their new permanent head coach.
Burnley manager Kompany, Mauricio Pochettino and one other unnamed manager are being considered by the Premier League club.
However, ex-Bayern Munich boss Julian Naglesmann is no longer in the running to take over at Stamford Bridge.
Chelsea sporting directors Laurence Stewart and Paul Winstanley are leading the recruitment process.
The Blues sacked manager Graham Potter at the start of April after less than seven months in charge, following a 2-0 home defeat by Aston Villa that left the club 11th in the Premier League table.
Ex-Chelsea boss Frank Lampard was then named interim manager until the end of the season.
Former Manchester City captain Kompany has this season led Burnley to promotion from the Championship in his first year in charge at Turf Moor.
The 37-year-old Belgian has transformed the team's style resulting in attractive, attacking football which has been praised by supporters and those within the game.
It would be a significant jump for Kompany, who has only managed Anderlecht (2019-2022) before his appointment at Burnley.
Kompany's impressive season has also led to him being linked with the vacant Tottenham job.
Chelsea had already held talks with German Nagelsmann, who was sacked by Bayern in March.
But there was scepticism from the club after they received mixed feedback on Naglesmann during the talks process.
Meanwhile, Pochettino has been without a job since being sacked by Paris St-Germain last summer.
The 51-year-old Argentine is best known for his work at Chelsea's rivals Tottenham, where he managed for five years and reached the Champions League final in 2019.
BBC Sport understands there is currently no favoured choice at the club and the process is ongoing.
Former Spain and Barcelona manager Luis Enrique had been among the contenders but he has reportedly not made Chelsea's shortlist after initial talks.
Chelsea, who are 11th in the Premier League and out of all other competitions, next play on Wednesday, when they welcome Brentford to Stamford Bridge (19:45 BST).
'Kompany should be wary, Chelsea is toxic'
Former West Ham, Aston Villa and Bolton midfielder Nigel Reo-Coker told BBC Radio 5 Live that Kompany being on the shortlist was a "surprise to me".
"Are they interested in him because they've seen what he's done at Burnley, they've studied it and they think he's the right man for what they need?" asked Reo-Coker.
"Or is it again that Todd Boehly has seen something shiny and that everyone is talking about him and thought 'let's go for him'?
"That's the problem at Chelsea, I don't see good leadership and them going after managers for the right reason and right purpose.
"If I was Vincent Kompany I'd be flattered but wary of going because right now, for me, that club is toxic."
Don Hutchison, who made 220 Premier League appearances for five clubs, thinks Chelsea should be considering Brighton boss Roberto di Zerbi.
"I don't think he [Boehly] should look any further than Roberto di Zerbi," he told BBC Radio 5 Live.
"Brighton gave them an absolute footballing lesson last week. Chelsea have gone soft and I've never associated that with them.
"Chelsea fans and Todd Boehly might be thinking 'we can't go back to Brighton again' after what happened with Graham Potter but actually be courageous, if you have to go back to Brighton to get the right man then be courageous and put your neck on the line and get the best manager for the job.
"If you bring De Zerbi in, give him money, give him licence to bring players in, but also choose the 12-15 that go out the door."