The English Premier League side needs to bolster its forward line following the departure of Chris Wood to Newcastle United and has money to spend this month.
Iranian international Azmoun is in the final six months of his contract at Zenit St Petersburg and has scored 10 times in 21 games this season.
The 27-year-old has attracted interest from across Europe and it's believed Burnley has tabled a £8million bid for the player, lancs.live reported.
Zenit prepares to accept that figure but with leading continental clubs eyeing a move, Burnley faces a tough task in signing the former Rubin Kazan striker.