Gunn impressed during his spell on loan with Norwich City in the second tier last season, leading to numerous top-flight clubs taking an interest.
Southampton have now got a deal over the line for Gunn, and the young keeper is already eyeing the number-one jersey at St Mary's Stadium.
"I'm coming in and I'm really excited. I want to push for the number-one spot straightaway," he told the club's official website. "I've come to play Premier League football and I'll be doing everything in my power in training to get that.
"Obviously, in the future, anything can happen, and I'm really positive about what the club holds."