Alonso has 12 months left on his current Madrid deal, and the Spanish giants are keen to keep him in the Spanish capital.
Because Mourinho, who was unveiled as the new Chelsea boss earlier this month, has been tipped to return to Spain with an offer for Alonso.
The World Cup winner was a fixture in Mourinho's starting lineup last season, making 28 La Liga appearances, providing five assists.
Sources in Spain claim Real Madrid president Florentino Perez has held talks with Alonso's agents regarding a contract extension but nothing has been agreed so far.
The Spaniard moved to Real Madrid from Liverpool in a £30m deal in the summer of 2009.
He's won a La Liga title in the Spanish capital, and is widely regarded as one of the finest deep lying midfielders in the world.
Frank Lampard recently agreed a new one-year contract extension at Stamford Bridge, but the Chelsea legend can no longer play every game.
John Obi Mikel usually operates in a similar position when fit, but Mourinho reportedly sees Alonso as the perfect midfielder for the role and wants him to join him in west London.
But the report claims Madrid are keen to prevent this, and will redouble their efforts to tie Alonso down to a new long-term deal.
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