Despite speculation linking the Sporting Lisbon left back to the likes of Liverpool and Chelsea, the Saints have stolen a march on their Premier League rivals for the 24-year-old.
The Sporting Lisbon left back impressed in Argentina's run to the World Cup final, performing well in a defense that only conceded four goals all tournament.
Rojo is primarily a fullback but can also operate in central defense and it is understood he will cost the Saints in the region of 15 million euros.
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Koeman is keen on the 24-year-old following Luke Shaw's move to Manchester United, while Dejan Lovren's expected departure to Liverpool edges ever closer.
Rojo has spent two years at Sporting, after moving to the Portuguese club in 2012 from Spartak Moscow and has two years remaining on his current contract.
He has been heavily linked with a move to the Premier League this summer, with Liverpool and Chelsea keen on his services but Goal understands the Saints have moved to the head of the queue.