The Spaniard's current club, Real Madrid, could be willing to listen to offers for the 32-year-old, amid concerns the player has lost too much weight, say reports in Spain.
According to El Confidencial, Real are concerned by the Spain star's lack of fitness and apparent weight loss during pre-season, with Madrid boss Carlo Ancelotti also worried by Alonso showing 'little strength' during training.
Alonso joined Liverpool from Real Sociedad in 2004, and went on to score 19 goals in 210 appearances, including the equalising strike in the Reds famous Champions League success against AC Milan in 2005.
The player joined Madrid in 2009, winning several further honours and taking his tally of international caps to an impressive 113. But following the arrival of Toni Kroos at the Bernabeu this summer, his place in the Real team is severely under threat.
With Madrid also believed to be interested in signing Brazilian star Luis Gustavo from Wolfsburg, Alonso could find himself moving on this month. There is little doubt that he would be welcomed back with open arms at Liverpool, but any such deal is merely speculation at this stage.