Athletic Bilbao and Inter Milan have reportedly made enquiries about the midfielder, with Bilbao having made an offer between £20 and 25 million, according to Spanish newspaper Marca.
The defensive midfielder was signed from Real Sociedad in 2013 for £25 million in 2013, but has struggled to establish himself at the Bernabeu. Now he has fallen further down the pecking order after the arrival of German midfielder Toni Kroos in the summer.
So far he has started 18 La Liga games in as many months with the club, and that has sparked rumours that manager Carlo Ancellotti is willing to let the Spaniard leave in the summer.
Arsenal manager Arsene Wenger is said to be a long-term admirer of the deep lying midfielder, who was supposed to be the long-term successor to Xabi Alonso in the Madrid midfield. That position is one the Gunners desperately need to fill, while it has been suggested Liverpool may look to him as the replacement for USA-bound Steven Gerrard.
However, the English duo have reportedly fallen behind Inter Milan and Bilbao in the race for his signature, with both clubs keen on a deal. It is thought that the pair think they have a good chance of agreeing a loan deal, with an option to buy at the end of it.