Richard Ratcliffe called on the Government to insist that Nazanin Zaghari-Ratcliffe was immediately tested for the virus by officials at the Evin prison in Tehran.
Although there are currently no confirmed cases at the prison, Covid-19 has spread rapidly across Iran – with at least 43 dead amid 593 patients identified.
On Sunday the Foreign Office announced it is pulling all non-essential staff out of its embassy in Tehran due to increasing concerns about the spread of coronavirus in Iran.
In an update on Saturday, issued through the Free Nazanin Campaign, Mr Ratcliffe said his wife had seen "no improvement" and was suffering with a continual cold sweat and a feeling of nausea.
In a phone call with her family, Mrs Zaghari-Ratcliffe said: "For a long time this has not felt like a normal cold.
"These symptoms have lasted almost a week. I know I need to get medicine to get better. This does not go magically."
Mr Ratcliffe said his wife had informed the prison guards of her symptoms and that she suspected she could be suffering from coronavirus, but she is yet to be tested.
Following reports of conditions at the prison on Friday, he said a new batch of disinfectants, gloves and masks were made available to the ward.
"We are concerned by the prison authorities’ refusal to test her and the wider suppression of coronavirus inside the Iranian prison system," Mr Ratcliffe said.
"We call on the UK Government to insist that Nazanin is tested for coronavirus immediately and is treated properly.
"We further call on the Prime Minister to take charge as part of his coronavirus efforts to ensure that British Iranians held hostage in Evin prison are diplomatically protected."
On Thursday, Commons leader Jacob Rees-Mogg said that while the Iranian authorities have denied any outbreak of coronavirus at the prison, he would "not always take as authoritative" such remarks.
Mrs Zaghari-Ratcliffe was arrested at Tehran’s Imam Khomeini airport while travelling to show her young daughter, Gabriella, to her parents in April 2016.
She was sentenced to five years in prison over allegations, which she denies, of plotting to overthrow the Tehran government.
She was later afforded diplomatic protection by the UK Government, which argues that she is innocent and that her treatment by Iran failed to meet obligations under international law.
The Foreign and Commonwealth Office (FCO) called on Iran "immediately allow" health officials into a prison in Tehran where Nazanin Zaghari-Ratcliffe believes she has contracted coronavirus.
Her husband Richard Ratcliffe had earlier raised concerns at Evin prison's "refusal to test her" for the virus, which has spread rapidly in Iran.
A Foreign Office spokesman said: "We are urgently seeking information from the Iranian authorities on reports that coronavirus is spreading in Evin prison, including to British-Iranian dual nationals.
"We call on the Iranian government to immediately allow health professionals into Evin prison to assess the situation of British-Iranian dual nationals there."