The facility - which is celebrating its centenary this year - is one of Cheltenham's most famous buildings, but desperately needs to expand.
The competition brief, launched in conjunction with Cheltenham Borough Council - calls for a design that 'will sit within the footprint of a disparate collection of buildings and an adjacent semi-derelict site'.
The new facilities must also provide 'larger, flexible temporary exhibition spaces; better public circulation and access within the galleries; improved storage, including greater public access to study collections; and dedicated space for outreach and other development work'.
The brief also says the winning design must be 'constructed and run in an environmentally friendly and innovative way, using reclaimed materials and on-site/off-site sustainable energy sources'.
CBC said the competition will be a flagship for its public 'green' scheme and added that the scheme would enhance one of the town's significant conservation areas, which also includes Cheltenham's oldest building, the 13th-century St Mary's Church.
Museum and Arts Manager, Jane Lillystone, said: 'We are looking for an outstanding design which is unique, functional, eco-friendly and achievable. We are delighted to be launching this competition during Architecture Week, particularly given this year's theme of 'How Green is Our Space?' and look forward to receiving submissions over the coming weeks and months.'
For more details of the competition click here: www.cheltenham.artgallery.museum/buildingforanewfuture or contact Joanne Wallis, Competitions Officer, RIBA Competitions Office on 0113 234 1335 or firstname.lastname@example.org.