The Spain international is reportedly locked in contract negotiations with the La Liga champions, but no agreement has been reached, leading Alonso to consider his options elsewhere.
The 30-year-old is thought to have enjoyed his time at Liverpool, and is believed to be open to a Premier League return.
Alonso is under contract at the Bernabeu until 2014, but Real Madrid are refusing to continue talks over a new deal before the end of the season.
And the Spanish champions are reportedly taking a hardline approach to contract renewal negotiations with several of their big-name stars. Angel Di Maria, Raul Albiol and another former-Liverpool star Alvaro Arbeloa are all being made to wait by Real Madrid.
And the Mirror reports the stalling tactic could result in Alonso leaving the Spanish giants, with his advisers thought to be exploring a possible return to England.
Alonso joined Liverpool for around £10m from Real Sociedad in 2004, and went on to make almost 150 Premier League appearances for the Reds.
The World Cup winner spent five years on Merseyside, and quickly established himself as a fan favourite at Anfield under Rafael Benitez.
But the 30-year-old swapped Liverpool for Real Madrid in 2009, linking up with new Bernabeu boss Jose Mourinho after the Spanish giants shelled out close to £30m for the deep lying playmaker.
In a Twitter question and answer session conducted by Adidas today, Liverpool captain Steven Gerrard revealed Xabi Alonso is the one Liverpool player he would love to bring back to Anfield, saying he's the best midfielder he has played with.
Alonso and Real Madrid face Borussia Dortmund in the Champions League on Tuesday evening at the Bernabeu. Following their defeat in Germany in the last matchweek, Madrid need to get a result against Jurgen Klopp's side to give themselves the best chance to progress to the knock-out round.
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