According to the Daily Express, Brendan Rodgers is ready to spend more than Rojo's asking price to ensure that Liverpool is the only suitor. United and Arsenal have also been heavily linked with the Argentina international.
The 24-year-old can play at left-back or centre-half and featured in Argentina's recent 2-1 win over Bosnia and Herzegovina at the World Cup last weekend.
Rodgers values his versatility and Liverpool have a real need to strengthen in defence, especially at left-back. Jon Flanagan deputised at the position for large parts of last season, and while he performed well, Rodgers wants a more established name next season as they prepare for the Champions League.
As well as Liverpool, United, Arsenal, PSG and Monaco have all been mentioned as possible destinations for the young defender, who signed for Sporting Lisbon from Spartak Moscow two years ago.
Sporting paid £3.5m to bring him in from Russia, and the report claims they want at least £12m now. But Liverpool are prepared to pay more than that to ensure they get their man.
Rodgers' search for a left-back has dragged on now for several months. The Anfield manager thought he had secured a deal to bring in Spanish international Alberto Moreno from Sevilla.
A move for the teenager is likely too expensive for Liverpool, with Rojo more within their price range as Rodgers looks to spread his transfer budget across several targets.
Rojo played in an attacking wing-back position during Argentina's first group game in Brazil, and flashed his defensive instincts and impressive technical ability on several occasions. We're not sure how happy Rodgers would be with this though...
His form in Portugal is what has brought him to the attention of the likes of United and Liverpool, and Sporting must hope that this tournament will only drive his valuation higher.
Argentina have a simple group, and having dispatched their main rival for top spot in their maiden game, they must be extremely confident of progression to the latter stages of the competition.