A new leader for Rutland County Council will be elected at a meeting on 26 March.
A £6m project would see Rutland County Museum extended to house the world-famous ichthyosaur.
The Conservative group had brought the motion that was discussed at a special meeting on Tuesday.
The head of the authority could go in a dispute over the handling of government reforms.
Two representatives say they saw work being carried out without cones around the affected area.
Campaigners say the budget just approved offers no assurances about the future of their library.
Ashley Baxter, leader of South Kesteven District Council, suggests merging with Rutland.
Councillors have said an independent review into the land will take place in due course.
England's smallest county could be merged with larger neighbours under reorganisation plans.
