The petition is set to be handed to Dudley Council tonight by members of the town's civic society.
The council's cabinet will be discussing £20 million worth of savings that have been earmarked for the next three years, which include shutting the museum.
The cabinet is expected to advise the programme of cuts be approved by the full council.
Finance boss Councillor David Sparks has said leaders must take action to avoid the authority going bankrupt and have blamed Government cuts for the dire financial situation it currently finds itself in.
Axing up to 275 jobs, raising council tax by almost two per cent and cutting opening hours at libraries also form part of the plans.
But it is the proposal to shut the museum and art gallery that has proved to be the most controversial. The total that had signed the petition last night stood at 2,430*.