Christopher Jones died earlier this month aged 56, and is believed to have lived in Oakengates since about 2001.
The coroner's office and West Mercia Police are appealing to find Mr Jones's next of kin following "extensive enquiries".
The authorities believe he was born in Wolverhampton and may have a sister called Linda who used to live in Woodside, Telford.
A police spokesperson said: "Can you help the coroner for Shropshire identify the next of kin for a man who died earlier this month?
"Despite extensive enquiries, no next of kin has been traced for Christopher Jones, aged 56.
"Mr Jones resided in Oakengates, Shropshire from around 2001.
"We believe he was born in Wolverhampton and he may have a sister called Linda, who in 2001 was possibly living at Wolverley Court, Woodside.
"If anyone knows anything about Linda, or any other family for Christopher, please can you make contact with the coroner’s office.
"The office is unmanned from December 25 - 29, but messages can be left."
To contact the coroner's office, call 01743258540.