Blues chief Todd Boehly and his board, including co-owner Behdad Eghbali, sacked Graham Potter at the start of April.
Chelsea legend and former boss Frank Lampard was then installed as caretaker until the end of the season.
However, even the icon’s return to the dug-out has failed to provide a lift as he has suffered three consecutive defeats since taking over, the latest being Saturday’s 2-1 home loss to Brighton.
Boehly has held a number of talks already with potential successors to Potter and former Barcelona and Spain supremo Luis Enrique emerged as an early frontrunner to take charge.
He held a video call with Chelsea chiefs just days after Potter’s sacking and then flew into London for face-to-face talks.
German Julian Nagelsmann is also in the mix along with ex-Tottenham manager Mauricio Pochettino.
Napoli’s Luciano Spalletti and Sporting Lisbon’s Ruben Amorim are others believed to have been spoken to by Chelsea chiefs.
But now a surprise new name has been linked with the vacancy by The Times in the City legend.
They claim that the Blues “also admire” Kompany after the job he has done at Burnley.
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The 37-year-old Belgian took charge of the Clarets following their relegation from the Premier League last summer and has led Burnley straight back into the top-flight in his first season at the helm.
SunSport told you last month the former defender was a surprise name on Daniel Levy’s shortlist for the Tottenham job.
And he has also been touted to replace Pep Guardiola at Man City, who he was captain of under the Spaniard before becoming Anderlecht manager in 2019.
He won two of his four Premier League crowns with Guardiola.
But Boehly has been impressed by Kompany’s work at Turf Moor and who he has transformed the club around in such a short space of time.
Now it seems that he would like the chance to talk with him about trying to perform a similar job in West London.