Celtic, Leeds United and Wolverhampton Wanderers have all been linked with moves for the 24-year-old.
Pope is thought to be on several clubs' radar
And despite the likelihood of his path to the first team being blocked by the 30-year-old initially, Pope is keen on the move to Turf Moor, according to the Sun.
The presence of England's Heaton could limit Pope's game time at Burnley
It was reported by Teamtalk last month that Celtic and Leeds United were among the many sides chasing the former Bury Town shot-stopper, while Wolves were also listed as suitors by South London Today.
However, at least two of those avenues look to have closed, with Wolves bringing in Fulham number 1 Andy Lonergan on Wednesday, and Leeds set to extend the contract of young prospect Bailey Peacock-Farrell.
Lonergan arrived at Molineux from Fulham this week
And Celtic are also fairly well-stocked in goal, with internationals Craig Gordon and Logan Bailly already on the Bhoys books.
Scotland's Gordon is part of a healthy Bhoys goalkeeping contingent