The 25-year-old becomes the club's third signing of the summer and the second of last year's loanees to make a permanent switch after Rory Holden.
He joined Southampton in 2017, two years before his loan move to the Banks's Stadium.
“I'm really pleased to come back," said Rose.
“I spoke to the manager and Maik (Taylor) throughout the lockdown and during the off season and I'm really happy to get it all sorted.
“I was here last season and I really enjoyed my time here and I'm excited to get going.
“For myself, it is easy coming back here because I know all the lads and it won't be difficult for me to settle into the team.
“The one point for me will be to try and play as many games as I can. Robbo and I push each other on and it is good to have competition."