Two female, ex-Wikileaks volunteers accused Mr Assange, an Australian citizen, of sexually exploiting them while he was in Stockholm to give a lecture.
Assange assert the sex was consensual and the accusations are politically aggravated.
Asked earlier this week if he felt solidarity with the WikiLeaks founder, Ecuador’s leftist President told a TV interviewer “of course, but we also feel solidarity for England and for English and international law.”
At a press conference on Wednesday, Patiño released details of a letter he said was delivered through a British embassy official in Quito, the capital of the South American country.
The letter said, “You need to be aware that there is a legal base in the UK, the Diplomatic and Consular Premises Act 1987,that would allow us to take actions in order to arrest Mr Assange in the current premises of the embassy.”
It added, “We need to reiterate that we consider the continued use of the diplomatic premises in this way incompatible with the Vienna convention and unsustainable and we have made clear the serious implications that this has for our diplomatic relations.”
In June, judges at the UK's Supreme Court dismissed his final appeal against banishment to Sweden.
An offer by Ecuador to allow Swedish investigators to interview Assange inside the embassy was rejected.