Swedish authorities want to question the Australian founder of the whistle-blowing website over accusations of rape and sexual assault made by two female former WikiLeaks volunteers during a visit there in August 2010.
Assange, who has been living in Britain since his arrest here in December last year, denies wrongdoing.
In his appeal to be heard on Monday (local time), he is seeking to have his case referred to Britain's Supreme Court. A decision could come soon after.
If the appeal fails, Assange may face extradition within 10 days.
The 40-year-old spent nine days in London's Wandsworth prison after his arrest last year. He was freed a week before Christmas on bail and has since been living at the country house of a wealthy supporter in eastern England.
His arrest came shortly after WikiLeaks published thousands of secret U.S. diplomatic cables that included unflattering views of world leaders and candid assessments of security threats.
Assange says the allegations are politically motivated.
Assange lost his last attempt to avoid being sent to Sweden on November 2 after two High Court judges upheld a previous ruling.
The application to take the case to Britain's Supreme Court rests on two legal questions: is the warrant for Assange's arrest valid, and can he be considered an "accused" person as required under extradition laws when no decision has been taken over whether he will be prosecuted.
If his appeal fails, Assange could take his case to the European Court of Human Rights in Strasbourg, further prolonging his stay in Britain.
In 2010, WikiLeaks posted 391,832 secret papers on the Iraq war and 77,000 classified Pentagon documents on the Afghan conflict. It has also made available about 250,000 individual cables, daily traffic between the State Department and more than 270 American diplomatic outposts around the world.
