On Monday judges at the High Court in London said that his fight against deportation to face rape allegations in Sweden raised an issue of public importance, and ruled that he could ask the Supreme Court to consider the case.
The 40-year-old Australian now has until December 19 to lodge his application with the Supreme Court -- although there is no guarantee that they will agree to take on the case.
"We have informed the parties that if they get the papers in by the 19th then we will make a decision before Christmas," Supreme Court spokeswoman Sian Lewis told AFP.
She said there would not be a hearing but that a panel of three Supreme Court judges would examine Assange's application on paper, before deciding whether to formally listen to his appeal at a later date.
If they refuse to hear the case then he will be extradited within 10 days.
Assange was arrested exactly one year ago on a European arrest warrant issued by Sweden after allegations by two women of sexual assault and rape.
At the brief High Court hearing on Monday, two judges said there was a case of "general public importance" raised by Assange's lawyers about whether a Swedish prosecutor was a proper judicial authority to sign the warrant.
Assange, who is living at a friend's mansion in eastern England under strict bail conditions, strongly denies any wrongdoing and says the sex with the women in Sweden was consensual.
He has said the allegations are politically motivated and linked to WikiLeaks' release of hundreds of thousands of classified US files about the wars in Iraq and Afghanistan.