Clarets boss Kompany is keen to continue to add to his squad at Turf Moor as the new season draws closer and his eyes have focused on an option in France.
He has zeroed in on Metz and their Mali international defender Kouyate.
Burnley are looking to thrash out a deal for the 25-year-old and it has been suggested they are prepared to pay around £2.5m to take him to England.
Kouyate is claimed to have been eyeing a move to play his football in England, which should help Burnley’s cause.
He was signed by Metz from fellow French side Troyes in 2020.
Kouyate, who goes by his nickname Kiki, made 30 appearances in Ligue 1 for Metz last season, picking up eight yellow cards in the process.
He could not help Metz avoid relegation and the club slid down into Ligue 2 after finishing second bottom.
Kouyate has a further two years left on his deal at Metz.