Mark Law, 42, went missing from Arbroath but is known to often visit Edinburgh, leading to police enquiries in the capital.
He was last seen in North Grimsby, Arbroath, on Thursday morning, at around 8.30am.
Officers are also searching in Angus, as Mr Law is known to also have links there.
He is described as white, around 5 ft 9 inches tall, with short brown hair.
When last seen he was wearing a green tracksuit, a blue gilet and a dark coloured beanie hat.
Police said Mr Law takes daily medication, and it is not known whether or not he had this with him when he was last seen.
Police Scotland said in a statement: “Police Scotland is appealing for information that could help trace a 42-year-old, man missing from Arbroath. Mark Law was last seen around 8.25am in North Grimsby, Arbroath on Thursday, 6 May, 2021.
“Mark takes daily medication and it is unknown if he has his medication with him. He has not been seen or heard from since Thursday and concern for his well-being is growing.
“Mark has links to the Angus area and often visits Edinburgh. Officers are carrying out enquiries in those areas in an effort to locate Mark and ensure he is ok.
“Anyone who may have seen Mark or have any information or knowledge as to his whereabouts is asked to call Police Scotland via 101, quoting incident 1096 of 8 May, 2021.”