Snibston Discovery Museum, which is based at a former colliery, showcases the area's industrial heritage.
But the withdrawal of a £4m Heritage Lottery Fund bid has prompted fears within the museum community that the centre could be in danger.
The county council said the funding came with strings attached and Snibston was part of a wider review of museums.
The award-winning museum is the most visited in the county, but some of the buildings were included on English Heritage's most recent At Risk register.
The funding bid had been for a £5m project called Mining Lives, which would have improved the condition of the colliery buildings and increased public access to the site.
Mark Taylor, from the Museums Association, said: "I just hope they have thought this through and they have left room to change their minds.
"Because it seems to me that when you are going to get £5m to improve your flagship museum it breaks your heart if you have to send that back, especially if the real reason is that you want to close that museum."
The council, which needs to save to save £95m, said a review of the county's museums was taking place but there were no plans to close the centre.
A Heritage Lottery Fund statement said: "As well as providing support to address the extensive repair needs of the scheduled ancient monument, the project would have created much-needed training opportunities for local people.
"The decision not to take this project forward is therefore very disappointing."
It added: "Our concern now is that the colliery buildings remain at risk, and we look forward to hearing from Leicestershire County Council what plans it has for the future of Snibston and the Scheduled Ancient Monument."
The county council's deputy leader, Nick Rushton, said accepting the money would bind them to conditions which meant they would be unable to properly review ways of saving money on site.
"I just dare not take this £4m from the Heritage Lottery Fund at this moment in time," he said.
"I have a big review ongoing at the moment of all arts, libraries and museums. I have to guarantee to protect vulnerable adults, children and elderly persons and I have to save £95m."

Bethan Rhys Roberts hosts Ask the Leader on BBC One Wales at 19:00 BST on Wednesday.
Kirsty Williams has been a member of the Welsh Assembly since it was created in 1999 - at 28, she was one of the youngest AMs elected.
After the 2007 election, she successfully opposed her party entering into a so-called rainbow coalition with the Conservatives and Plaid Cymru in Cardiff Bay.
The episode caused divisions in the party - which she ended up leading some 18 months later.
Early on in her leadership Ms Williams launched Project 31 - a plan which would see the Welsh Liberal Democrats aim to win a majority of seats in Cardiff Bay.
However, in her only assembly election campaign in charge so far - 2011 - the Lib Dems lost one of their six seats, and polls suggest they have a fight to keep the rest this time around.
Despite leading the smallest party in the assembly, Ms Williams can take credit for getting Welsh Labour ministers to deliver Lib Dem policies such as increased school funding for poorer pupils as part of a deal to support the government's budget.
Her reputation as a party leader may be enhanced - but the biggest challenge for Kirsty Williams is making sure that there are enough Lib Dem AMs re-elected to ensure that she has a party group to lead for the next five years.
Thursday 14 April - Plaid Cymru leader Leanne Wood in Aberystwyth
Friday 15 April - Welsh Labour leader Carwyn Jones in Llangollen
Monday 11 April - Welsh Conservative leader Andrew RT Davies in Swansea
Tuesday 12 April - UKIP Wales leader Nathan Gill in Swansea
Ask the Leader can be seen on BBC One Wales at 19:00 each day and on the BBC iPlayer.
You can also follow the programmes via social media - @walespolitics
The future of one of Leicestershire's main museums has been questioned after an application for funding was shelved.
[NEXT_CONCEPT]
Welsh Liberal Democrat leader Kirsty Williams faces questions from an audience in Aberystwyth in the third of a daily series of live TV election specials.