The left back, who impressed at the World Cup, is also attracting interest from Manchester United and Liverpool but the Saints are keen to trump other potential suitors.
Goal reported Ronald Koeman's initial interest in the Sporting Lisbon defender – who can also play at center back – earlier in July as the club looks for a replacement for Luke Shaw.
Rojo himself has revealed he could be open to any move if the price is right, giving Southampton some hope that it could persuade the World Cup star to switch to St. Mary's.
"Everyone has a price," said 24-year-old Rojo. "If a good deal arrives, then I’m sure I will sit down with the club and evaluate things."
Southampton is trying to push through a quick deal for Rojo in order to fend of interest from elsewhere in the Premier League.
Manchester United, Chelsea, Liverpool and Tottenham are all keeping tabs on Rojo's situation but the south-cost side is keen to better other potential suitors.
There is also interest from La Liga and the Bundesliga, but a move to England is Rojo's priority according to sources close to the player.