The Sun says they will offer to double Pope's salary to keep him at Turf Moor.
Goalkeeper Pope is on around £15,000 a week after signing as cover for Tom Heaton.
But the England call-up has shone this term since Heaton dislocated a shoulder.
The Clarets face a dilemma with their No 1 back to full fitness and Newcastle United heading the Pope fan club.
So boss Sean Dyche will offer the 25-year-old a bumper new deal and then decide on Heaton's future.
As the better-paid keeper, Heaton, 31, may end up being the one to leave the club this summer.