The Australian has spent more than three years in Belmarsh prison in London after he was dragged out of the Ecuadorian embassy where he had been living since 2012.
The High Court in London confirmed to the BBC an application from Mr Assange has been submitted.
He is wanted by American authorities over documents, relating to the Iraq and Afghanistan wars, which were leaked in 2010 and 2011 and which the US says broke the law and endangered lives.
Mr Assange had until Friday to decide whether or not to appeal against his extradition.
His wife Stella previously pledged to fight the decision with “every available avenue”, saying: “I’m going to use every waking hour fighting for Julian until he is free.”
She said her husband had told her “recently” he planned to kill himself if he was extradited.
His lawyer Jennifer Robinson has said his appeal would go “all the way – if necessary to the European Court of Human Rights”.
At the time the extradition decision was made, a Home Office spokesperson said: “Under the Extradition Act 2003, the Secretary of State must sign an extradition order if there are no grounds to prohibit the order being made. Extradition requests are only sent to the Home Secretary once a judge decides it can proceed after considering various aspects of the case.
“On 17 June, following consideration by both the Magistrates’ Court and High Court, the extradition of Mr Julian Assange to the US was ordered.”
They added that “UK courts have not found that it would be oppressive, unjust or an abuse of process to extradite Mr Assange.
“Nor have they found that extradition would be incompatible with his human rights, including his right to a fair trial and to freedom of expression, and that whilst in the US he will be treated appropriately, including in relation to his health.”