The 31-year-old midfielder was a big fan favourite at Anfield during his five years on Merseyside. He moved to Real Madrid in 2009, and has gone on to become a key fixture in the club's starting eleven.
But he's been ruled out for up to three months this term after suffering a fractured metatarsal in his right foot in preseason.
He underwent a successful operation in August and hopes to return in November.
But speculation suggests Real Madrid could be willing to offload the Spain international in January after securing his replacement - Asier Illarramendi was signed this summer for €38m from Real Sociedad. Luka Modric has also proved himself to be an able deputy, forming a strong partnership with Sami Khedira.
And Liverpool, along with three Serie A clubs - Fiorentina, Juventus and Napoli - are believed to be keen on the Spaniard.
Liverpool and Napoli are believed to be leading the race - Liverpool because of Alonso's public affection for the club and Napoli because they're now managed by Alonso's former-Anfield boss Rafael Benitez.
He's still a hero at Anfield after helping the club to win the 2005 Champions League, but Fiorentina - flush with cash following the summer sale of Stevan Jovetic - and long-term admirers Juve are also thought to be monitoring developments.
Liverpool have started the season in excellent form, and currently sit top of the Premier League table after three wins and a draw from four games.