Cook was a stand-out performer for Bournemouth this season and his efforts have not gone unnoticed.
The 23-year-old central defender has established himself as one of the best young players outside the Premier League.
A number of clubs have been credited with holding an interest in Cook, but Burnley are hoping to steal a march on their rivals by luring him to Turf Moor.
Burnley boss Sean Dyche is actively looking to bolster his squad ahead of a return to the Premier League and is thought to have identified Cook as one of their top targets.
Bournemouth tied Cook down to a new deal last summer, but they are bracing themselves for offers for the former Brighton man as a result of his fine form at Dean Court.