Rafinha has yet to feature for Barcelona this season after suffering a knee injury last term.
The Brazilian featured 28 times under Luis Enrique last campaign, scoring seven goals.
Rafinha Alcantara, whose brother Thiago left Barcelona for Bayern Munich in 2013, may not be in the plans of new Barca boss Ernesto Valverde.
And Spanish transfer website Don Balon claim Real Madrid have passed up the opportunity to sign him on three occassions.
The first time was before his loan spell with Celta Vigo in the 2013/14 season.
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Real chief Perez has always been a fan of the Alcantara brothers, although it is said he was keener on playmaker Thiago.
The second time was during Rafinha’s time at Barcelona under Enrique, when he was left out of the team despite being promised game time.
That caused his father and agent Mazinho to offer him to Real Madrid, but again they refused.
And Don Balon say the third time came following the announcement of Valverde at the Nou Camp, with the message being Rafinha would not play a major part in his plans.
But despite being offered the 24-year-old through third parties Real again refused.