The United States has slapped 18 charges against the 47-year-old Australian, who had been hiding in London's Ecuadorian embassy for seven years until his arrest on April 12.
Sweden has since reopened a 2010 rape investigation against Assange and could file its own extradition request.
The Westminster Magistrates Court in London held a brief hearing on Thursday at which it moved the US case to June 12.
Chief magistrate Emma Arbuthnot referred to Assange as “not very well”, with the next hearing possibly taking place at the Belmarsh prison where he is serving a separate 50-week sentence for skipping bail.
“It may be more convenient for everyone if it's there,” Arbuthnot said.
WikiLeaks expressed “grave concerns” Wednesday over Assange's condition. The websites, which published a trove of classified State Department and Pentagon material about US operations in Afghanistan and Iraq, said Assange has been moved to the prison's health ward.
“During the seven weeks in Belmarsh his health has continued to deteriorate and he has dramatically lost weight,” WikiLeaks said in a statement. “The decision of prison authorities to move him to the health ward speaks for itself.”
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Around 30 supporters, some wearing Assange masks and one dressed as a giant whistle to represent his whistleblowing activities, chanted “we want justice” and “stop killing Assange” outside the courthouse.”Don't shoot the messenger,” one placard said.
Sixteen of the 17 US charges under the Espionage Act are related to obtaining and disseminating classified information. A separate count covers his alleged hacking activities.
The military documents and diplomatic cables were obtained by former US Army intelligence analyst Chelsea Manning, who was sentenced in 2013 under the Espionage Act to 35 years in prison over the leaks.
Assange could be sentenced to 175 years in prison if convicted on all 18 counts.
The case has upset US and other world media, which argue that Assange's activities differ very little from their own. They fear his case could set a precedent for limiting free speech and media rights. Washington accuses Assange of putting the lives of US agents, diplomats and informants at risk.
The ultimate decision of whether to extradite him to the United States or Sweden would rest with the UK interior ministry.