Conservative group leader Thomas Kerr has written to the city treasurer calling for a council tax 'moratorium'.
The public health approach comes as UK and Scottish governments prepare to host drug summits.
Both events will discuss Scotland's drugs crisis in Glasgow later this month.
Britain officially split from the European Union at 11pm, three-and-a-half years since the Brexit vote.
Constitutional relations secretary Mike Russell will take part in talks with UK Government ministers and counterparts from the Welsh administration.