Hart lost his place in the England team during a loan spell with West Ham United last season, being left out of the World Cup squad as Jordan Pickford established himself as the number one.
However, the 31-year-old has been in good form since joining Burnley during the summer, making a string of saves despite the Clarets still waiting for their first win of the new season.
The Mirror reports that Southgate will attend Burnley's Premier League clash with Bournemouth at Turf Moor, where he will be able to watch the likes of Hart, James Tarkowski, Callum Wilson and Lewis Cook.
Southgate has already suggested that Jack Butland's place in the England squad is under threat due to the fact that he plays in the Championship, while uncapped duo Marcus Bettinelli and Alex McCarthy were called up to the last England squad.
Hart has not featured for England since last November, when he played in a friendly against Brazil.