The 31-year-old is understood to have rejected the offer of a new contract at the Bernabeu, in the hope of making a dramatic return to the Premier League next season.
However Fiorentina will look to tempt the Spanish World Cup winner to Serie A with a handsome contract offer.
Liverpool will be desperate to complete the signing and bring a true legend back to Anfield.
Alonso was instrumental for the Reds after arriving from Real Sociedad for £10.5million in 2004.
During his first season at Anfield, Alonso played his part in the club’s famous Champions League triumph over AC Milan, scoring the goal to make it 3-3 and force extra time and penalties on that famous night in Istanbul in 2005.
A year later Alonso was the man in the middle as Liverpool claimed the FA Cup courtesy of a dramatic final win against West Ham.
The Spaniard would go on to prove crucial over the next few seasons also, as Liverpool narrowly missed out on both the Champions League and Premier League title under Rafael Benitez, before joining Real for £30million in 2009.
While Liverpool will remain favourites to bring a club legend home, Fiorentina have taken confidence from his comments made earlier this summer when questioned on a possible Premier League return.
“The Premier League? No,” Alonso said.
“No way, no chance.”
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