According to The Sun, the 25-year-old could be tempted by the move if he loses his place in the Clarets first team to Tom Heaton, who is on the brink of returning from injury.
Burnley are said to be keen on retaining the services of both goalkeepers, but the report suggests they may not have a say in the matter, unless they can guarantee Pope first-team football.
Heaton was in strong form for Burnley and even broke into the England setup before a shoulder injury sidelined him last September.
Newcastle, meanwhile, also have the option to sign goalkeeper Martin Dubravka on a permanent deal for £3.5m this summer.
The Slovakia international, who is on loan at the Magpies from Sparta Prague, made his Premier League debut during Newcastle's 1-0 win over Manchester United last weekend.