There are fears that the team leader job at The Lillie Art Gallery in Milngavie is under threat.
East Dunbartonshire Leisure and Culture Trust (EDLC) is currently carrying out a service review across libraries and art galleries to cut costs.
Staff have heard that one option being discussed is to get rid of team leaders - however they are currently in charge of more than one gallery or museum.
The team leader for The Lillie Art Gallery is also team leader for the Auld Kirk Museum, in Kirkintilloch.
An EDLC member of staff, who asked not to be named, said: “There are only two full-time members of staff at The Lillie Art Gallery and it’s the same at the Auld Kirk.
“I don’t see how they’d manage with just one full-time member of staff.
“The team leader organises exhibitions, and the other member of staff is mainly at the reception desk.”
Mark Grant, General Manager of EDLC Trust, said: “EDLC’s structure is currently under review across all service areas. At this stage of the review no decisions have been taken so we are unable to comment until the review process has been completed.”