The 25-year-old has spent two seasons at the Enschede-based outfit, scoring 28 goals in 66 appearances since moving to the club from Eredivisie rivals Groningen.
There are no details regarding a fee for the Serbia international who has scored six goals in 25 appearances for his country.
Tadic has been earmarked by Southampton’s new manager Ronald Koeman as a replacement for Lallana, who completed his move to Liverpool earlier on Tuesday.
Tadic’s international team-mate Lazar Markovic is also thought to be a target for Liverpool and has been linked with a move to Anfield from Benfica.
Southampton are also understood to be keen on reinforcing their front-line following Rickie Lambert’s departure and have been linked with a raid on Koeman's old club Feyenoord for their Italian striker Graziano Pelle.
You'll be able to watch action from the Eredivisie next season on Sky Sports 5.