Pope has enjoyed an impressive season between the sticks at Turf Moor, despite the Clarets struggling for Premier League survival.
Burnley host Newcastle United on the final day of the season on Sunday, requiring to at least match Leeds United's result at Brentford to retain their top-flight status.
However, whether Burnley avoid the drop or not, Pope is said to have numerous suitors ahead of the upcoming transfer window, with The Sun reporting that both West Ham United and newly-promoted Fulham are keen on the shot stopper.
The duo have been told to stump up £40m if they wish to take the England international away from Turf Moor though, which is double what both were hoping to pay according to the report.
Since making the £1m move from Charlton Athletic back in 2016, Pope has gone on to make 154 appearances for Burnley across all competitions, as well as earning eight caps for England during his stay so far.