The initial call to emergency services raised concerns about the welfare of a homeless person who had been seen collapsed.
But police enquiries have now established that the 56-year-old, who also went by the surname Swan, lived in Prosper Street, Hunslet.
Her death is not being treated as suspicious and the Coroner’s Office has been informed.
Detective Inspector James Entwistle, of Leeds District CID, said: “We have been making extensive enquiries to trace Fiona’s next of kin to inform them of her death but we have so far been unable to identify any.
“We always endeavour to make direct contact with relatives ourselves but we have been left with no other option than to make a public appeal to find them.
"There has been a suggestion that she has a sister in London but we have no further information about that. We would ask that any relatives of Fiona or anyone who has any information that could assist us gets in touch.
“We have had some names put forward for the three people that were with her in the time leading up to her death but we are yet to trace them and we would still like to speak to them.”
Anyone with information is asked to contact Leeds District CID via 101, quoting log number 326 of January 8.