The Danish striker is set to be released by the Gunners in the coming weeks after 10 years at the club, with Arsene Wenger having lost patience with his below par performances and off-field problems.
This has opened the door for Clarets boss Sean Dyche, who is keen to add the experienced goal-getter to his squad as they plot to stay in the big time.
Bendtner is also wanted by Russian side Rubin Kazan and Crystal Palace, yet Burnley are hopeful that they cam beat the competition to snap the 26-year-old.
The Scandinavian ace is thought to be upset at his release from Arsenal, and could be tempted to stay in English football to prove his worth to his old club.
However, the former Juventus loanee may opt to wait it out and assess his options this summer in the hope of securing a big move.
Bendtner found himself in hot water at Arsenal in recent months after going AWOL in Copenhagen, where he got into trouble with the police on a night out.