Saints have been trying to sign the Bologna striker for a number of weeks now in a deal which could reach a transfer fee of around £11m.
The Uruguayan is thought to be keen on a move to St Mary's after refusing to play for the Italian outfit in order to force through the switch, and Adkins says that the club are working hard to land the player.
"Until the pieces of paper are signed you can never say it's done but the chairman is working exceptionally hard behind scenes and is backing us in what we do," said Adkins.
"We are very ambitious and there is serious cash to be outlaid if it's going to happen."
The 21-year-old has nine caps for his country.