The Mail on Sunday says the Italian giants have tried to play down the significance of Beckham's loan spell to avoid further angering LA Galaxy.
Milan vice-president Adriano Galliani even claimed the club do not want Beckham beyond March.
But a source said: "Forget the public posturing - if David does well in his loan period until March, Milan will try to keep him until the end of the season. Whether or not they are successful will depend on Beckham and his relationship with the Galaxy.
"No one is talking about it in public right now because if they did, it would cause trouble between the clubs."
Beckham is meant to rejoin LA Galaxy by April 15, when the US transfer window closes, but could return earlier.