However, his agent Matteo Roggi insisted: "There is no chance of him leaving in January. He wants to stay and try as hard as possible not to be relegated. He has never been relegated with any team, so that would be terrible for him and West Ham."
"It is too early to say what will happen with Paolo's future but I'm quite sure he does not feel he will return to Italy," said Roggi.
"His family speak English better than they do Italian. He is more English now than Italian."
Di Canio was injured during the Hammers 1-0 defeat against Southampton on Monday night and now faces at least five weeks on the sidelines following a cartilage operation.
"The injury does not affect his future situation," said Roggi. "We are not speaking about him renewing his contract with West Ham at the moment. We are going to see what is going to happen in January.
"Paolo would really like stay at West Ham and finish his career. We had a talk with them to see what they wanted to do, but they say they are not ready to speak about it."
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