Almost a year after his arrest over rape and sexual assault allegations, the40-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 the former hacker lost his first appeal against deportation.
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 WikiLeaks boss 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 last year.
Assange has spent much of the last year under virtual house arrest on a supporter's country estate in eastern England after a European Arrest Warrant was issued last December.
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.
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.
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.
Support for the eccentric, platinum blond WikiLeaks chief has dwindled amid a slew 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.