Since moving to St Mary's for an initial £10m in 2018, the former Manchester City stopper has made 30 appearances in all competitions.
However, after conceding nine goals against Leicester City in October, the 24-year-old has been restricted to three appearances in the FA Cup.
According to The Sun, the Saints are willing to part ways with Gunn ahead of next season.
While the Scottish outfit want to re-sign the former England stopper, a deal will only take place if Southampton cannot find a buyer for Gunn. body check tags ::