Coventry Transport Museum jobs under threat
All staff at Coventry Transport Museum have been offered voluntary redundancy as part of £100,000 savings.
Culture Coventry, which runs the attraction, said incomes had dropped in the last year due to fewer admissions.
While entry to the main exhibits is free, it said the turnover in the cafe and shop had been affected.
Chief Executive Gary Hall said it was not yet clear how many posts needed to be cut, but a number of people had already come forward.
The museum employs more than 80 staff, as well as volunteers, and has not ruled out compulsory redundancies.
Mr Hall said the museum needed to "balance the books" and said new staff terms and conditions had already been introduced "to protect jobs as far as possible".
The new terms include withdrawing extra rates of pay for weekend shifts.
Culture Coventry is also responsible for managing the Herbert Art Gallery and Museum, Lunt Roman Fort and the Priory Visitors Centre.