The 31-year-old has so far failed to negotiate a new contract to extend his stay at the Santiago Bernabeu, which could force Jose Mourinho to sell the Spanish international when the transfer window reopens in the New Year.
Reds boss Brendan Rodgers would likely welcome the opportunity to bring Alonso back to Anfield, reuniting him in the middle of the park with club captain Steven Gerrard.
Alonso enjoyed five memorable seasons on Merseyside, and was at the centre of some of Liverpool's greatest triumphs, including their epic Champions League final victory over AC Milan in Istanbul in 2005.
The Real Madrid star made a total of 121 appearances for Liverpool before returning to his homeland in 2009, and has always spoken affectionately of his former employers, saying the club will forever hold a special place in his heart.
However, in light of these reports linking him with an emotional return to England, Sky Sports presenter and Spanish football correspondent Guillem Balague has rubbished claims Alonso will rejoin Liverpool, but didn't deny that he could leave Real Madrid in January.
Despite playing a pivotal role in the club's title-winning campaign, as Mourinho's men successfully wrestled the La Liga crown from Barcelona's grasp last term, his future is now shrouded in doubt as contract talks have reached an impasse.