The pensioner lived at the McCarthy and Stone Summerfield Place development on Wenlock Road and died over the weekend after testing positive for Covid-19 on Friday.
It is understood to have been the first coronavirus death in Shropshire and was confirmed as Boris Johnson said all people in the UK should avoid pubs, clubs and theatres, stop all non-essential contact and travel, and work from home if they can.
A spokesman for McCarthy and Stone said the resident had underlying health issues.
“On Friday one of the residents at Summerfield Place tested positive for Covid-19," they said.
"They had underlying health issues and sadly passed away at the weekend.
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"We have offered our deepest condolences to the family. We’re not in a position to provide further details and the family have requested privacy at this time.
"We are providing full support to them and the other residents at the development.
On Monday the first coronavirus death in Wales was confirmed at Wrexham Maelor Hospital, meaning at least 36 people have died in the UK after being diagnosed with coronavirus - not including the Shrewsbury death.
Meanwhile the University Centre in Shrewsbury closed its campus after a student showed symptoms and Boris Johnson was set to hold a televised press conference on Monday following an emergency meeting of the Cobra committee.
Across the UK, 1,543 people have now tested positive for Covid-19, up from 1,372 at the same time on Sunday.
However, many thousands more are estimated to have been infected and local authority figures are no longer being released due to the fact people with mild symptoms are no longer being tested.
Education Secretary Gavin Williamson is meeting school leaders on Monday to discuss the implications of school closures and exams being postponed, while Environment Secretary George Eustice is holding a conference call with food suppliers and supermarkets regarding continuity of supply.
There has also been confusion over exactly what elderly people will be asked to do going forward, after Mr Hancock said on Sunday they will be asked to stay at home for a very long time.
About 170,000 people have been infected by Covid-19 across the world and more than 6,500 have died.
The McCarthy and Stone spokesman said the safety and wellbeing of residents was the company's absolute priority and it had been monitoring international and local developments very closely.
“Our residents choose us because they want to live in their own homes, and we support them to live independently.
"While we are not a care home, we are doing all we possibly can to ensure our residents and staff have the support and services they need during this period.
"We are very much aware of the potential vulnerability of our residents and are committed to taking every step within our power to minimise the risk of Covid-19.
"In addition, we are strongly promoting the advice and guidance issued by the government and Public Health England.”