Having seen a £10m offer rejected, the Saints have tabled an improved bid and are growing increasingly confident of landing the midfielder from Feyenoord.
Some finer details of the deal still need to be finalised, but Sportsmail understands a deal is close to completion.
Clasie was linked with Southampton after rumours emerged about Morgan Schneiderlin’s move to Manchester United.
The midfielder’s £25m transfer to Old Trafford paved the way for Saints manager Ronald Koeman to move for Clasie, and it is poised to be announced by the club.
Newcastle United were also interested in the 24-year-old, but Southampton appear to have all but beat them to his signature.