Martinez is expected to be sold by Real at the end of the season, with Barcelona seeing the centre-half as a natural replacement for Carles Puyol.
Real Madrid have also been monitoring his performances this term.
Meanwhile, Spanish sources say City boss Manuel Pellegrini wants to bring Martinez to England, where he would arrive as a direct replacement for Bosman prospect Joleon Lescott.
Martinez, 22, carries a €30 million buyout clause in his current contract.
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