Mr Hamill, 39, was last seen in his home in the Dumbarton Road area at around 2.10am on Monday, May 10, and he has not been heard from since.
He is described as white, 5ft 10in tall and of medium build.
He has short, fair hair but it is not known what clothes he was wearing when he went missing.
Concerns are growing for Mr Hamill’s welfare, but he is known to have links to Kinghorn in Fife and it is possible that he may have travelled there.
Inspector Katrina McLaren from Partick Police station said: “Along with his family, we are becoming increasingly concerned for Brian’s welfare and we are keen to find him as soon as possible.
“We have a number of police resources out looking for Brian, we know that he does have links to the Kinghorn area in Fife so we are also asking anyone from the area if they have any information which might help us find him.
“I would urge anyone who may have seen Brian to contact us. Similarly, Brian if you are reading this, let us know that you are safe and well.”
If you have any information you should contact Police Scotland on 101, quoting 2969 of May 10, 2021.