(CNN) -- Top diplomats from Ecuador and the United Kingdom failed to reach an agreement Thursday over the case of WikiLeaks founder Julian Assange, but they agreed to meet again to discuss the issue.
"We still do not see, of course, an immediate solution, but we understand that there is a willingness to discuss the topic." Ecuadorian Foreign Minister Ricardo Patino told reporters after the meeting on the sidelines of the United Nations General Assembly in New York.
Ecuador granted Assange asylum in August, but he faces arrest in Britain if he leaves the South American country's embassy in London.
On Thursday, British Foreign Secretary William Hague told Patino "that the UK was under an obligation to extradite Mr. Assange to Sweden" and that "the concept of 'diplomatic asylum,' while well-established in Latin America, did not feature in UK law," a spokesman for his office said.
For much of 2011 until June, Assange had been under house arrest in Britain while he filed appeals against his extradition from Britain to Sweden. Assange has not been charged with a crime, but Sweden has said it wants to question Assange on allegations that he sexually assaulted a woman there.
Assange has denied the sex allegations, describing them as a ruse to get him to Sweden, which would then extradite him to the United States. Several U.S. officials have asserted that Assange violated the law by publishing in 2010 and 2011 a trove of classified war documents and diplomatic cables.
Patino called on British authorities to reconsider the case, given Ecuador's granting of asylum and "the right to not be extradited when politically motivated causes are involved."
The Ecuadorian Embassy is a sovereign space upon which authorities from other countries cannot encroach.
But Patino told reporters Thursday that a key concern for Ecuador is that it is currently impossible for Assange "to enjoy his condition of asylum" while forced to stay inside the embassy.
Ecuadorian authorities were also concerned about what would happen if Assange faced a medical emergency inside the embassy, Patino said.
"Both ministers agreed that they were committed to the search for a diplomatic solution to Mr. Assange's case," the UK foreign secretary's spokesman said.
The founder of WikiLeaks delivered an impassioned appeal Wednesday for the U.S. government to end its actions against him, his website and those who support it.
"It is time for the United States to cease its persecution of WikiLeaks, to cease its persecution of our people and to cease its persecution of our alleged sources," Assange, speaking via satellite from London, told a packed conference room at the United Nations, where world leaders were attending the United Nations General Assembly.
The event was held by the Mission of Ecuador on U.N. grounds, but was not officially sponsored by the world body.
CNN's Ashley Fantz contributed to this report.