The England international has agreed a four-year contract with the former Premier League champions in a deal worth a reported £30m.
Reed told the club's official website: "While we are sad to see Luke depart the club, we fought to ensure that we got the right deal for a player in whom we have invested a great deal of work over the past decade.
"Luke joined us at the age of eight, and our academy staff and coaches have helped him to progress into a top-class player. He will leave our club and carry forward our top-class pedigree, as others have done before.
"This deal is a good one for Southampton Football Club, and hopefully sends a clear message to other clubs wishing to bid for our players. Luke's transfer fee will be reinvested into the team as our new manager, Ronald Koeman, builds for the new season and for years to come."