The England midfielder, on loan at Milan until March 9, desperately wants to stay with the Serie A club and increase his chances of playing at a fourth World Cup in 2010.
Major League Soccer said a deal had to be done by last Friday but Milan and Beckham's advisors have virtually ignored the deadline and are confident the transfer can still take place.
"Up to March 9, I think there is time to find a solution," Milan coach Carlo Ancelotti told reporters after Sunday's 2-1 derby defeat by leaders Inter Milan.
Beckham has fought tough battles before and won.
He virtually took England to the 2002 World Cup single-handedly and forced Fabio Capello to bring him back into the title-winning Real Madrid side in 2007 after being dropped when he announced he would be leaving for Galaxy.
It looked unlikely two years ago that he would equal Bobby Moore's England outfield caps record of 108 but he managed it in last week's defeat by Spain having burst back on to the scene.
Beckham's options are varied and it is unclear which road he will take, but he is very likely to end up at Milan.
His preference is for the clubs to agree a deal, which could take place outside of the transfer window because he is technically on loan at Milan until June.
Talks will take place this week but the two parties are poles part when it comes to a valuation.
"ALWAYS PROFESSIONAL"
Beckham has done well but Milan do not want to pay a lot for a 33-year-old, who will probably only be given an 18-month deal.
Various figures have been bandied about in the media after Milan's opening offer of 5 million euros was turned down. Some reports have said Galaxy want around 12 million while Milan are only willing to go to 8 or even 10.
Another possibility is that Milan do a swap deal with one of their players or use their contacts to help Galaxy recruit another European name from elsewhere.
Milan defender Paolo Maldini is due to retire at the end of the season and having spent much of his recent leisure time in the U.S, it might not be unreasonable to speculate that he could be interested.
Beckham could buy out his Galaxy contract or use a get-out clause leave for free in November.
He is a very rich man but how much he would have to pay is confidential. Given the five-year deal he signed with Galaxy was worth around $250 million, it would not be a small sum.
Beckham is disillusioned by the standard of U.S. soccer but buying out his contract is not the image he wants to purvey.
"I'm always professional so we'll have to wait and see when the final decision is made," Beckham told reporters on Sunday when asked if he would go back to Galaxy if he had to.
"Talks have obviously been going on and on and everyone's been talking about it," he added with rolling eyes.
Milan must also be tired of the negotiations given they have come quickly after the Kaka saga, where Manchester City were willing to pay a world record
