Attorney Jennifer Robinson, who has represented Assange, said it wasn't clear whether Assange would try to get the European Court involved.
"(That's) still being considered," she said in a tweet.
Assange denies any wrongdoing in the Swedish case, saying that the sex was consensual. He and WikiLeaks' supporters have suggested the case was being manipulated to political ends by foes angered over WikiLeaks' revelations. The website has been responsible for the largest leaks of classified documents in U.S. history, among them some 250,000 U.S. State Department cables.
Britain's Supreme Court has given Assange a two week-long grace period before extradition proceedings can start. Once the two-week period is up, officials have 10 days to fly Assange to Sweden. If the European Court does not intervene, Assange could be sent out of Britain between June 28 and July 7.
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Associated Press Writer Louise Nordstrom in Stockholm, Sweden contributed to this report.