The Saints want to strengthen their defence, as they have seen Mohammed Salisu and Duje Caleta-Car leave and Armel Bella-Kotchap is attracting interest from Borussia Dortmund and Bayern Munich.
Russell Martin is keen on Swansea man Wood and is pushing for a deal.
Southampton had open talks with Swansea regarding a deal but failed with their initial bid.
The Saints have submitted a revised bid to Swansea worth up to £10m including add-ons.
It has been claimed that the improved bid consists of more achievable targets for add-ons, while paying £7m upfront.
The 21-year-old centre-back is keen on a reunion with his former Swansea boss Martin.
Martin brought in Woods last season from Middlesbrough and the player featured 40 times in the league for the Swans.
Now it remains to be seen whether Southampton’s new bid will be able to convince Swansea to let Woods go this summer.
