Alonso only signed a new deal with the La Liga giants in January but there are claims Los Blancos are prepared to sell the midfielder this summer.
It has been suggested Real boss Carlo Ancelotti is unhappy with Alonso's poor condition on his return from the World Cup in Brazil and, as such, the 32-year-old could be on the move, with Toni Kroos providing extra competition in central midfield following his summer move from Bayern Munich.
Liverpool have been linked with Alonso on a number of occasions since his departure for the Bernabeu in 2009 and they could be tempted to make a move now Real are open to a sale.
Meanwhile, keeper Pepe Reina is set to leave Liverpool in an initial £2million move to German giants Bayern Munich.
Reina spent last season on loan at Napoli and had been linked with a return to Serie A but Liverpool have now confirmed they have agreed to sell the Spaniard to the Bundesliga champions.