The man, thought to be aged 39, was reported missing from the village of Auchterhouse, Tayside, on Thursday, February 26.
He was last seen at the beginning of the month in Montrose.
Officers believe him to be called Andrew McLeod, but said they had not been able to confirm that.
House-to-house enquiries in Auchterhouse have also failed to provide leads, where locals have told police they do not know the man - or recognise his photograph.
Now officers are widening the parameters of their search in case the man lives elsewhere in Scotland.
Investigators are also checking relevant CCTV images for any additional information they might provide.
Launching a nationwide appeal, Police Scotland said they had been informed that the missing man may have been suffering recently with personal issues.
Inspector Martin Pattie said: “Concern has been raised for this person’s welfare and our priority is to find him. Despite our efforts, we have been unable to trace him so far.
“We need to establish the man’s identity and alert his family and friends to his disappearance.”
He added: “I would urge the man to contact us so we know he is safe and well.
“I am also appealing to anyone who recognises the man in the photograph to get in touch.”
Anyone with information can call Police Scotland on 101, quoting incident number 1280 of February 26, 2021.
UPDATE: Police Scotland has confirmed the man in question has since been traced.
A statement read: “The man was reported missing in good faith and we have now traced him and he is safe and well.
“Enquiries established that the photograph, name and age in the appeal was not a true identity.”