Manager Jose Mourinho wanted to sign Tevez in January and the Argentina international remains his top target.
Tevez signed a five-year deal at Eastlands in 2009 but handed in a transfer request earlier this season.
Transfer tussle: Manchester City face a battle to keep Carlos Tevez
The striker admitted he was unhappy spending long periods of time away from his daughters in Argentina and his relationship with boss Roberto Mancini has never been a smooth one.
According to the Daily Mirror, Real general manager Jorge Valdano has approached Kia Joorabchian, who represents the player, about a deal.
Forward thinking: Jose Mourinho wants to bring in a big-name striker
Madrid are desperate to land a big name as they look to close the gap on Spanish champions elect Barcelona and Mourinho has always been an admirer of Tevez, who has also been linked with Inter Milan.
Tevez, 27, has 19 Premier League goals for City this season, with former Manchester United team-mate Dimitar Berbatov the only player ahead of him in the table.?
City play Tottenham tonight (Tuesday) and Stoke in the FA Cup final on Saturday but Tevez is facing a race to be fit as he recovers from a hamstring problem.
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?Explore more:People: Carlos Tevez, Dimitar Berbatov, Roberto Mancini, Jose Mourinho Places: Madrid, Barcelona, Argentina
Source: Daily Mail
Source: Daily Mail