Lenin Moreno, the country's president, is in London for a disabilities summit, but is allegedly in discussions with British officials over a deal to hand the Australian over to police.
Mr Assange, who has been described as an "inherited problem" by Mr Moreno, could lose his diplomatic protection in a matter of days, The Intercept reported.
The 47-year-old has been living in Ecuador's embassy in London since June 2012, when he successfully sought asylum to avoid extradition to Sweden to face questioning about allegations of sex crimes he has always denied.
Those allegations have since been dropped, but Mr Assange would be arrested by British police should he leave the embassy for breaching bail conditions.
He believes that would pave the way for extradition to the US for the publication of a huge cache of American diplomatic and military secrets on the WikiLeaks website.
Speculation about Mr Assange's future has grown this month after the Sunday Times said senior officials from Ecuador and Britain have been in discussions since last week about how to remove him from the embassy after revocation of his asylum.
"The situation is very serious. Things are coming to a head," the source, who spoke on condition on anonymity, told Reuters. He said the latest information from inside the embassy was, "It's not looking good".
However, both the Ecuadorean government and British government sources played down suggestions there was likely to be any imminent movement to break the stalemate.
"The Ecuadorian state will only talk and promote understandings about Mr Assange's asylum, within the framework of international law, with the interested party's lawyers and with the British government," Ecuador's foreign ministry said in a statement ahead of the visit.
"At the moment, due to the complexity of the topic, a short or long-term solution is not in sight."
A British government source also said there was no sign of immediate progress. Last month, Foreign Office minister Alan Duncan told parliament that they were increasingly concerned about Mr Assange's health.
"It is our wish that this is brought to an end, and we would like to make the assurance that if he were to step out of the embassy, he would be treated humanely and properly," Mr Duncan said. "The first priority would be to look after his health, which we think is deteriorating."
WikiLeaks itself has added to speculation its editor could be facing arrest after its Twitter account retweeted multiple news outlets reporting his likely eviction.
It comes after 12 Russian spies were charged with election interference in the US. They allegedly hacked the emails of Hillary Clinton and the Democratic National Committee and passed them to WikiLeaks, which published them during the 2016 US presidential election.
In May, Mr Moreno ordered the withdrawal of additional security from Ecuador's small diplomatic headquarters in London. That followed a decision in March to suspend Mr Assange's communication system and direct voice to the outside world.
Mr Assange's supporters regard him as a champion of free speech who has exposed government abuses of power at great personal cost. His critics see him as a criminal who recklessly endangered lives in many countries by exposing secrets.
His attempt to halt British legal proceedings failed in February when a judge said Mr Assange appeared to consider himself above the normal rules of law.
Britain ended a permanent guard outside the embassy in October 2015 after spending almost £13 million in policing, but has vowed he will be arrested should he leave the embassy.
A US Department of Justice investigation into WikiLeaks is continuing but a US law official close to the investigation told Reuters there was no sealed indictment against Mr Assange.
Mr Assange has said the US would unveil charges against him if he were arrested by British police and this would pave the way for his extradition.
The Foreign Office declined to comment.
Additional reporting by Reuters
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