The Dutch international midfielder has emerged as the Saints’ main target to replace Morgan Schneiderlin, with the Frenchman having joined Premier League rivals Manchester United on Monday.
Southampton opened talks with Eredivisie outfit Feyenoord over the weekend, but their initial bid has been rejected by the Rotterdam-based outfit.
They want around £13million for 24-year-old Clasie, a product of their academy system, but the offer put to them was nearer £10million.
However, a source has told HITC Sport that Southampton – under the guidance of former Feyenoord boss Ronald Koeman - are confident of getting their man.
"Although the first bid was rejected, talks are ongoing and there is a willingness from both sides to see this finalised," the source told HITC Sport.
The Saints will be hoping that a compromise can be reached as they find themselves in a similar situation to last summer, with key men – Schneiderlin, Nathaniel Clyne – leaving and Koeman forced to rebuild during the summer window.