The 25-year-old goalkeeper spent last season on loan at Stoke City, where he featured 15 times in the Championship.
Gunn recently met with Andrew Sparkes, the club’s goalkeeping coach, and Matt Crocker, Southampton’s director of football operations, to discuss his future.
The talks were said to be positive but it has been claimed that Southampton are preparing to look at moving him on and selling the goalkeeper this summer.
And newly-promoted Norwich are showing an interest in getting their hands on Gunn in the next transfer window.
The Canaries were interested in signing Gunn on a permanent deal following his loan stint at Carrow Road in the 2017/18 season but Southampton beat them to his signature.
Daniel Farke wants him again this summer and Norwich are considering taking Gunn to the club in the coming weeks.
Southampton may be unlikely to recoup the £10m they spent on him a few years ago, but are still keen to sell him.
A fee of around £5m is likely to be Gunn’s price tag this summer.
