The court said in its decision a petition from Dinah Rose, a lawyer representing Assange, "is without merit and it is dismissed," The Guardian reported.
Assange is being sought under a European Arrest Warrant in connection with allegations of sexual assault and rape in Sweden. He has denied the allegations.
In its 5-2 May decision, the Supreme Court rejected Assange's claim the European Arrest Warrant issued for him was invalid because it was made by a prosecutor, not a judge.
The court found "judicial authority" could mean a prosecutor besides a judge.
The Guardian said Assange's lawyers could appeal Thursday's decision to the European Court of Human Rights in Strasbourg, France, or agree to the extradition.
Assange's lawyers argued the court's May decision had been based on a legal point that was not argued in court, the BBC reported.
The Supreme Court said Thursday extradition proceedings could not begin for at least two weeks.
Swedish prosecutors have not filed charges against Assange, 40, but want to question him about allegations of sexual assault and rape made by two former WikiLeaks volunteers in 2010.
Assange, whose whistle-blowing Web site has posted thousands of American diplomatic cables and sensitive military files, said the sex was consensual and the allegations against him were politically motivated, the BBC said.