43-year-old Assange is unable to get hospital treatment because he fears he will be arrested by police if he leaves the embassy, The Telegraph reported, citing his supporters.
Assange, whose website WikiLeaks published classified US military and diplomatic documents, has been living inside the Ecuadorian embassy building here, since being granted diplomatic asylum by the Latin American nation in August 2012.
Britain wants to extradite him to Sweden under a European Arrest Warrant for questioning in relation to a sexual assault investigation.
Australia-born Assange fears that from Sweden he will be extradited to the US, where he could face 35 years in prison for publishing classified documents related to the Pentagon's activities in Iraq and Afghanistan on WikiLeaks.
Metropolitan police officers have been stationed outside the embassy since Assange entered the building and have been ordered to arrest him if he attempts to leave, the paper said.
"After two years unable to go outside living within the air-conditioned interior of the embassy, Assange is suffering from arrhythmia, which is a form of irregular heartbeat, a chronic cough and high blood pressure," the paper said, citing WikiLeaks sources.
The sources also said the lack of Vitamin D, which is produced by exposure to sunshine, is damaging his health and could lead to a host of conditions including asthma, diabetes, weak bones and even heightened risk of dementia.
The Ecuadorian embassy has asked the Foreign Office for permission to transport Assange to hospital in a diplomatic car but has received no response, it added.