Officers working on a missing person inquiry made the discovery in Market Street at about 23:55 on Wednesday.
A man has been detained and is helping police with their inquiries.
A Police Scotland spokesman said: "The investigation into the circumstances is in the very early stages of establishing who the remains belong to and how that person died."
He added: "Our inquiries have taken us to a number of addresses in the Montrose area where there is likely to be a police presence for some considerable time."
Police have found what are believed to be human remains at a house in Montrose.