The Sun on Sunday newspaper reported that PNE had enquired about a January loan for Heaton.
Aston Villa remain favourites to land the 32-year-old who was on standby for England's World Cup squad in the summer.
Heaton was Burnley's first-choice keeper until September 2017 when he suffered a serious shoulder injury.
He returned to fitness towards the end of last season but was unable to dislodge Nick Pope in the Clarets' goal.
This season, Joe Hart has been Burnley's No.1, with Heaton having to make do with a place on the bench.