The Spain midfielder looks increasingly close to sealing a move to the Bernabeu and was photographed at Madrid's Barajas airport on Monday evening, hours after flying into England following Liverpool's Asian tour, says the Daily Mail.
Madrid's long pursuit appears to have paid off, with reports in Spain suggesting Liverpool will accept less than £30million to sell both Alonso and right-back Alvaro Arbeloa to the club.
Rafa Benitez has grown tired of fielding questions about the transfer tug-of-war.
Spanish newspaper AS says the Liverpool boss took Alonso to one side before they left Singapore and told him: "I want you to resolve your situation as soon as possible, for better or worse.
"Ideally the transfer is resolved in 48 hours. I need some time to find a replacement."
After Alonso reiterated his desire to leave, Benitez reportedly said he would not stand in the way and Madrid are confident of signing the former Real Sociedad player on a four-year deal.