The 33-year-old, who has scored two goals in eight appearances for the Rossoneri, insisted that he has done all he can to stay in Italy and the deal is now in the hands of the two clubs.
"I have been playing well on the pitch and also saying that I want to come and I want to stay and that is all I can do," he said.
"It has been an incredible experience playing with some of these players and also playing at the San Siro, but I just hope it continues.
"There is still a few weeks left before the decision has to made and hopefully it is sorted out before them because so far I have been able to concentrate on my football and not worry about things that are going on off the field and I prefer to do that.
"We'll have to wait and see. These things take time sometimes and of course it's going to. It could go down to the last few days."
Milan coach Carlo Ancelotti has also expressed his desire to keep Beckham at the club.
"He has been a wonderful signing," Ancelotti said.
"He is a unique professional but if his stay at Milan only lasts until 9 March he will remain a beautiful memory for us."
With Beckham playing almost a full match on Sunday Ancelotti's side kept alive their faint hopes of catching city rivals Inter at the top of Serie A with a 1-0 win over Cagliari at the San Siro.
Dutch midfielder Clarence Seedorf exploited a defensive mix-up to tap into an empty net after 65 minutes.
The former England captain delivered another assured display before giving way to Luca Antonini in injury time.
Cagliari gave their hosts several uncomfortable moments early on with goalkeeper Christian Abbiati tested by efforts from Robert Aquafresca and Jeda in the first half.
At the other end Filippo Inzaghi had the ball in the net but saw his strike ruled out as he strayed offside in latching onto Seedorf's neat pass.
Andrea Cossu shuddered the woodwork of Abbiati's goal after the break with a fierce long-range shot, but the visitors' efforts were undone as Diego Lopez and keeper Federico Marchetti collided in pursuit of a loose ball and Seedorf profited.
The result keeps Milan third in the league, two points behind second-placed Juventus and 11 off Jose Mourinho's Inter.
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