Lawyer Gareth Peirce said he will ask senior judges in London Dec. 5 to certify that Assange's case be considered by the country's highest court, The Guardian reported Wednesday.
Among other things, Assange must establish that his case raises a question of law that is of general public importance, the newspaper said.
A Supreme Court hearing would be the third phase of Assange's appeal against being extradited to Sweden to face allegations of rape, sexual molestation and unlawful coercion by two women he met in Stockholm in August 2010. Assange has maintained his innocence.
Legal observers said Assange's decision could mean a verdict on whether he should be extradited may be delayed until as late as next summer, The Guardian said.