William Robinson, also known as Bob Dawson or Bob Roberts, is originally from the Bedfordhshire area but is believed to have travelled to the capital.
The 38-year-old was last seen in the Bedfordshire area on 13 April, however CCTV inquiries have shown him to have been in the Princes Street area on 14 and 22 April.
Officers are eager to trace William and ask anyone who has seen him, or knows of his current whereabouts, to come forward.
He is described as white, around 5ft 11ins tall and of stocky build. He has long dark hair and a beard. He is likely to be carrying a large rucksack and is possible wearing a blue or camouflage jacket and a multi-coloured hat.
Inspector Trish Robertson from Howdenhall Police Station said: “As the passage of time increases we are growing concerned for William’s welfare and would ask anyone who may have seen him in Edinburgh city centre, or the surrounding area, to contact us as soon as possible.
“Equally anyone who may have spoken to William, or knows of his current whereabouts, should contact officers immediately.”
Those with information should contact Police Scotland on 101, quoting incident number 1151 of 22 April.
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