The Australian-born activist is currently being held at the category A Belmarsh Prison in southeast London.
He is embroiled in a legal battle to avoid extradition to the United States, where he faces a prison sentence of up to 175 years on espionage charges.
Wife Stella Assange said she is "worried" about her husband and that the new few days would be "crucial" for his general health.
She said her husband had been feeling unwell last week but became ill on Friday, developing a cough and fever.
“He was given some paracetamol,” Ms Assange said. “He tested positive for Covid on Saturday, the same day thousands of people came out onto the streets to support him.
Ms Assange added: “I am obviously worried about him and the next few days will be crucial for his general health. He is now locked in his cell for 24 hours a day."
The Ministry of Justice has been approached for a comment.
Ms Assange added that she was overjoyed at the number of people who formed a human chain around parliament on Saturday, estimating there were well over 5,000 in attendance.
It was the biggest event of its kind in support of the WikiLeaks founder, who has won support from human rights organisations, journalist groups and others across the world.
Last month, US lawyers and journalists who visited Assange when he was at the Ecuadorian Embassy said they are suing the CIA, claiming it spied on their private conversations in violation of the Fourth Amendment.
The US has charged Assange with a total of 18 counts, some under the Espionage Act, and which in theory could carry a total of 175 years in jail.
They have accused him of publishing confidential military records and diplomatic cables, some of which were leaked to him by a former US army intelligence officer.
Mr Assange and his supporters say he was exposing what they believe were war crimes committed by the US and UK as part of the so-called war on terror.
He has received the support of journalism and press freedom organisations around the world, and a UN torture envoy has denounced the way he is being treated.
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