Juventus, Valencia and Benfica are also reported to be showing an interest, but it's Sunderland and Benfica who are thought to be the two frontrunners ready to battle it out for the youngster's services.
Gimenez was something of a revelation for Uruguay at the World Cup this summer, putting in an especially memorable performance against England as the South American side went on to win the game 2-1.
Despite impressing at the World Cup, the 6"1 defender arrived in Brazil off the back of a frustrating season at Madrid.
The youngster managed only one appearance in La Liga last term, and has found it tough to push his way into the first team with Diego Simeone very keen on the centre-back partnership of Miranda and Gimenez's Uruguay team-mate Diego Godin.
It's thought however that the Madrid boss is unwilling to sell the 19-year-old seeing him as a prospect for the future, but would consider letting the Uruguayan leave on loan to gain experience elsewhere.
The Black Cats are noticeably short of cover in the centre-back department, with back-up for Wes Brown and John O'Shea relatively unconvincing.
Valentin Roberge and Modibo Diakité both arrived at the Stadium of Light last summer, but have flopped on Wearside and continually struggled to make any sort of impact.
The Sunderland manager is likely to be in the market for another centre-back before the window closes at the end of this month, and a move for his Uruguayan compatriot Gimenez may prove a viable option.