Assange, 50, is wanted by US authorities on 18 counts, including a spying charge, relating to WikiLeaks' release of vast troves of confidential US military records and diplomatic cables which US officials have said put lives in danger.
Last month, Home Secretary Priti Patel approved his extradition, with her office saying UK courts had concluded his extradition would not be incompatible with his human rights, and that he would be treated appropriately.
Australian-born Assange's legal team have lodged an appeal against that decision at the High Court, his brother Gabriel Shipton confirmed.
The court must give its approval for the appeal to be heard but it is likely the legal case will take months to conclude.
"We also urge the Australian government to intervene immediately in the case to end this nightmare," Shipton told Reuters.
The saga began at the end of 2010 when Sweden sought Assange's extradition from the UK over allegations of sex crimes.
When he lost that case in 2012, he fled to the Ecuadorean embassy in London where he spent seven years.
When he was finally dragged out in April 2019, he was jailed for breaching UK bail conditions although the Swedish case against him had been dropped.
He has been fighting extradition to the United States since June 2019 and remains in jail.
"We're going to fight this. We're going to use every appeal avenue," his wife Stella Assange told reporters after Patel approved his extradition.
