The Wikileaks founder denies the claims and has said his European arrest warrant is "invalid and unenforceable".
In February, his lawyers told the Supreme Court judges that the Swedish prosecutor who had issued the warrant did not have the authority to do so.
The 40-year-old Australian has been on conditional bail in the UK.
Mr Assange is accused of raping one woman and "sexually molesting and coercing" another in Stockholm in August 2010 but he claims that the allegations against him are politically motivated.
Mr Assange's Wikileaks website published material from leaked diplomatic cables embarrassing several governments.
The key legal question for the seven judges is whether the prosecutor who issued the arrest warrant had the judicial authority to do so under provisions of the 2003 Extradition Act.
The court is due to deliver its judgement at 09:15 BST (08:15 GMT).
Further appeal?
At the February court hearing, Mr Assange's lawyer, Dinah Rose QC, argued that the Swedish prosecutor who had issued his warrant was a party in his case and was not therefore impartial or independent.
She also challenged whether a public prosecutor could be considered a "judicial authority" as required by the act.
Clare Montgomery QC, for the Swedish Prosecution Authority, argued that the High Court was plainly correct to accept that the term "judicial authority" had a wide meaning.
She said when the EAW framework had been set up, the drafters had intended it to include the prosecutors of many countries.
This "broader approach" recognised the "historic role" of public prosecutors within EU member states in authorising arrests and making extradition requests, she said.
If Mr Assange loses his Supreme Court appeal he could appeal to the European Court of Human Rights (ECHR).
If that court then decided to hear his challenge, Mr Assange could lodge an injunction to have the extradition process put on hold.
But the Crown Prosecution Service said if the ECHR declined to take the case "he will be extradited to Sweden as soon as arrangements can be made".
In February, his lawyers told the Supreme Court judges that the Swedish prosecutor who had issued the warrant did not have the authority to do so.
The 40-year-old Australian has been on conditional bail in the UK.
Mr Assange is accused of raping one woman and "sexually molesting and coercing" another in Stockholm in August 2010 but he claims that the allegations against him are politically motivated.
Mr Assange's Wikileaks website published material from leaked diplomatic cables embarrassing several governments.
The key legal question for the seven judges is whether the prosecutor who issued the arrest warrant had the judicial authority to do so under provisions of the 2003 Extradition Act.
The court is due to deliver its judgement at 09:15 BST (08:15 GMT).
Further appeal?
At the February court hearing, Mr Assange's lawyer, Dinah Rose QC, argued that the Swedish prosecutor who had issued his warrant was a party in his case and was not therefore impartial or independent.
She also challenged whether a public prosecutor could be considered a "judicial authority" as required by the act.
Clare Montgomery QC, for the Swedish Prosecution Authority, argued that the High Court was plainly correct to accept that the term "judicial authority" had a wide meaning.
She said when the EAW framework had been set up, the drafters had intended it to include the prosecutors of many countries.
This "broader approach" recognised the "historic role" of public prosecutors within EU member states in authorising arrests and making extradition requests, she said.
If Mr Assange loses his Supreme Court appeal he could appeal to the European Court of Human Rights (ECHR).
If that court then decided to hear his challenge, Mr Assange could lodge an injunction to have the extradition process put on hold.
But the Crown Prosecution Service said if the ECHR declined to take the case "he will be extradited to Sweden as soon as arrangements can be made".
