Both Premier League clubs are said to believe the player, who signed to the Spanish capital at 18 from Danubio, is an ideal candidate to bolster their rearguard ahead of next season.
Gimenez is one of the club’s four captains this season and his role has taken on even greater significance after several starts last summer, including his defensive partner Diego Godin.
The Uruguayan defender signed a new long-term contract with the Rojiblancos last summer until 2023 and the new deal includes a € 120million release clause.
The 25-year-old has been in Diego Simeone’s first team for six seasons and is a tough, aggressive, quick and imposing center-back.
However, Gimenez suffered 17 different injuries during his career, including 15 muscle injuries.