Assange, 40, has been on conditional bail and is sought to be extradited to Sweden to face allegation of sex offences, which he denies. His lawyers had argued before the Supreme Court that the European arrest warrant against him was "invalid".
The implementation of the May 30 judgement had been deferred for two weeks, during which Assange's lawyers were given time to seek reopening of the extradition case on the ground that the case had been decided on a point of law that was not argued by any side during the hearing.
The Supreme Court today confirmed that Assange's lawyers had asked to reopen the case. The court can decide not to review the case and such appeals are considered rare.
On May 30, Assange's lawyer, Dinah Rose, had sought two weeks to decide whether to ask the Supreme Court to reopen the case, which surprised legal observers.
Legal commentator Joshua Rozenberg had told BBC, "This is a very unusual thing. It's not happened since this court was set up. It happened in the Pinochet case in the House of Lords. Very unusual, and means there's everything left to play for still."
He added, "(He) can stay in this country for at least two weeks, while they consider making this unprecedented application to reopen the case on the basis that it was decided on a point of law in the Vienna Convention on the
Interpretation of Treaties that was simply not argued by either side and which the court gave no notice to either the crown prosecution service, representing the Swedish authorities, or Assange's lawyers, that they were considering taking into account."