Real's president, Florentino Perez confirmed his interest in the 27-year-old over the weekend saying the club "would love to have him", but he faces a tough task after Rafael Benitez made it abundantly clear he has no intention of letting his best players go this summer.
The club are already reported to have turned down "laughable" £25 million offers for Alonso, dismissing the bids as "nowhere near" their own valuation. But with just £20 million rumoured to be Benitez's budget for the summer, there may be no other alternative if he wants to bring in new blood.
Eager to begin talks, Perez told the Times: "Xabi is a magnificent Spanish player so of course we'd love to have him. But everything would involve reaching an agreement that is in the best interests of both clubs."
Bolstering his chances is news that Real have successfully re-signed striker Alvaro Negredo from Almeria and Perez plans to use him as a bargaining tool to tempt Benitez into a deal.
The 23-year-old, who Benitez has been monitoring for several months, would ordinarily have cost £12 million, but Real ensured a buy-back clause when they sold him in 2007 and have paid just £5m.
Should a player-plus-cash deal materialise, Benitez could land himself a consistent young striker and something in the region of £22 million.
This would bump the Liverpool transfer kitty up to around £40 million meaning the Benitez could get on with landing some of the names on his target list.
Having missed out on signing Gareth Barry, the Spaniard has been linked with PSG's Stephane Sessegnon, Valencia's David Silva, Middlesbrough's Stewart Downing and Carlos Tevez