The Saints are reportedly offering an initial £7m plus another £3m in performance-related add-ons.
Southampton boss Russell Martin is a huge fan of the 21-year-old, who has won one cap for the England under-21 national team.
Swansea are yet to respond to the proposal which would see them make a massive profit on the £250,000 they paid Middlesbrough to secure his services.
Wood made 40 appearances for Martin-led Swansea last season as they finished in the top half of the Championship rankings, just three points adrift of the play-off places.
The youngster was part of a defence that will feel they should have kept things tighter at the back after being breached a concerning 64 times, just four less than their goals scored tally (68).
To put their woeful defensive statistics into context, only four teams conceded more, with three of them suffering relegation to the English third-tier division.
If Southampton are successful in their approach, Martin will hope Wood can help his new side run a tight ship at the back, having already conceded six goals from his opening three games.
The Saints are looking to replace Mohammed Salisu, who completed a move to AS Monaco in a deal worth around £13m earlier this month.
The 24-year-old was one of the standout performers for Southampton last season in a campaign that ultimately ended in relegation for the south coast club.
He helped keep one clean sheet and averaged 2.3 tackles per 90 while completing 70% of his attempted dribbles.
The Ghanaian international was also a force to be reckoned with in air and ground duels after winning 61% of his total battles.
He would be difficult to replace, but Southampton will hope Wood can plug the gaping hole he left in the heart of the defence.
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