Almost a year after his arrest over claims of rape and sexual assault, the 40-year-old Australian will ask two judges at London's High Court to decide whether his appeal can proceed to the Supreme Court.
For the appeal to be heard in Britain's highest court, the judges must rule the case raises a question of general public importance.
Monday's hearing comes a month after his first appeal against a ruling that he can be sent to Sweden was rejected.
If the ruling goes against Assange, the British leg of his legal battle will end and he faces extradition to Sweden within 10 days, an outcome that would mark a new low for the former hacker after a string of controversies.
Swedish police want to quiz Assange over allegations made by two Swedish women of sex crimes, which he strongly denies.
Assange claims the allegations are politically motivated and linked to the activities of his anti-secrecy website, which angered the United States by publishing thousands of classified documents.
He has already appeared nine times in British courts since his detention under a European Arrest Warrant on December 7 last year.
He has spent much of the last year under virtual house arrest on a supporter's country estate in eastern England, where he has had to stick to strict bail conditions.
A lower court initially approved Assange's extradition in February, but he appealed to the High Court which rejected his challenge on November 2.
Assange made the application for the Supreme Court to hear the case one day before the legal time limit.
The judiciary of England and Wales said the High Court would "consider Julian Assange's application for a certificate of law of general public importance on 5 December."
Legal sources said a decision was expected on the same day but that it was not certain.
According to the website Sweden Vs. Assange, which supports his case, his legal team will question whether a European Arrest Warrant issued by a state prosecutor is valid.
They will also query whether he can be defined as "accused" despite having not been prosecuted, the website added.
Assange has previously expressed fears that his extradition to Sweden would lead to his transfer to the United States to face as yet unspecified charges of spying.
The legal challenge comes despite the fact that a Swedish public relations firm claimed last week that it had been hired by Assange and that he would soon be returning to Sweden to face questioning over the allegations.
Swedish prosecutors want to question Assange on suspicion of two counts of sexual molestation and an accusation of rape made by two Swedish women in August 2010.
If the High Court fails to back his appeal bid on Monday, it will be another blow for Assange after his fortunes deteriorated dramatically in the past year.
When he was arrested, WikiLeaks was riding high having just started releasing more than 250,000 classified US diplomatic cables, following the earlier publication of US military files on the war in Afghanistan.
But support for the eccentric, platinum blond WikiLeaks chief has faded amid a welter of controversies.
Former WikiLeaks colleagues have turned on him, attacking the way he ran the site. He also fell out with newspapers, including the New York Times and Britain's Guardian, that WikiLeaks initially worked with to release documents.
WikiLeaks' work has come under threat, with the site forced to suspend releasing files in October after a funding blockade.
It resumed publication Thursday, however, with the launch of a project on the global surveillance industry.