Ecuadorian officials say that they are "studying and analysing the request", after the UK Supreme Court rejected Assange's bid to reopen an appeal against extradition to Sweden over alleged sex crimes.
The UK Foreign Office said that as Mr Assange is now in diplomatic territory, he is beyond the reach of British police, but they are trying to resolve the situation as soon as possible.
Mr Assange fears that if he is removed back to Sweden, he will then be transferred to America, where he will face a jury for confidential information released by Wikileaks, and he could face the death penalty.
However, Swedish authorities say that the European Court of Human Rights (ECHR) would intervene if Mr Assange was to face the prospect of "inhuman or degrading treatment or an unfair trial" in the US.
Back in 2010, Mr Assange was briefly offered Ecuadorian citizenship after his Wikileaks website published a huge number of leaked diplomatic cables, to the intense embarrassment of many governments and international businesses.
Ecuador’s deputy foreign minister made the offer of citizenship, saying they wanted Mr Assange to be able to present his information freely. However, the country’s president, Rafael Correa, later dismissed the offer saying that he had not approved such an offer.
Sweden meanwhile want to question Mr Assange over claims from two women that he sexually assaulted them while he was in Stokholm back in mid-2010. No charges have yet been made.
Mr Assange has always claimed his innocence, and says the request is politically motivated.