The Saints have made the Argentine World Cup finalist their top target to replace the departed Luke Shaw, and are understood to have seen an initial offer of close to £9.5m knocked back.
Sporting want around £15m-£20m for their man, who is expected to leave this summer.
Liverpool and Chelsea have both been linked with him, yet their interest has cooled in recent times.
The Lisbon giants are aware of the multiple sides looking at their man, and are keen to use his impressive displays in Brazil to boost his value.
Cash should not be an issue for Southampton as it stands, with the sales of five key first-team players having brought in around £90m.
Some of this has been used to snap up Dusan Tadic and Graziano Pelle, but Ronald Koeman still has funds for big name arrivals.
Rojo is thought to be keen on a switch to the Premier League.