The Wikileaks co-founder has lived in its UK embassy since 2012 after seeking asylum to avoid extradition to Sweden over a rape claim - later dropped.
He was given a set of house rules by the London embassy this week, including taking better care of his cat.
Mr Assange faces arrest for allegedly breaching bail conditions if he leaves.
Wikileaks lawyer Baltasar Garzon is in Ecuador to launch the case, which the Press Association reports is expected to be heard in court next week.
Wikileaks said the country's government had threatened to remove the protection Mr Assange has had since being granted political asylum.
It added that his access to the outside world had been "summarily cut off".
In a memo, it threatened to confiscate the pet if he did not look after it, it said.
The embassy did remove Mr Assange's internet connection in March, accusing him of "interfering in other countries' affairs".
However, earlier this week it said it would be partially restored.
In a statement, Wikileaks said: "Ecuador's measures against Julian Assange have been widely condemned by the human rights community."
It claims the government of Ecuador refused a visit by Human Rights Watch general counsel Dinah PoKempner and had not allowed several meetings with his lawyers.
Mr Assange's lawyers also said they were challenging the legality of the Ecuador government's "special protocol" - which makes his political asylum dependant on "censoring" his freedom of opinion, speech and association.
Lawyers for Mr Assange claim the protocol:
- Requires journalists, Mr Assange's lawyers and anyone else looking to see him to disclose private or political details - such as the serial numbers and codes of their phones and tablets
- The protocol says the government may "share" the information "with other agencies"
- Allows the embassy to seize the property of Mr Assange or his visitors and, without a warrant, hand it over to UK authorities
Wikileaks said that US congressmen had written an open letter to Ecuador's president, Lenin Moreno, about the situation.
It claims the document said that in order to advance crucial matters - such as economic co-operation, counternarcotics assistance and the possible return of a USAID mission to Ecuador - they must first resolve a "significant challenge" - the status of Mr Assange.
More about: JulianAssange