Cornwall Council says it is committed to keeping the Royal Cornwall Museum open despite pulling its funding.
Earlier this week, the 200-year-old museum announced it might have to shut after its council funding was dropped.
The museum has received funding from the council for the past 50 years, previously getting around £79,500 a year.
More than 20 cultural organisations in the county got funding from the council's £1.8million Culture and Creative Investment fund - but this year the museum is not one of them.
Since the news was revealed there has been an outpouring of support for the museum, which says it has been overwhelmed by "incredible messages" from supporters.
Now Cornwall Councilhas issued a statement to say it acknowledges some applicants would be "disappointed" to be unsuccessful in their bid for council funding.
Cllr Carol Mould, portfolio holder for neighbourhoods, said: “The council has supported the Royal Cornwall Museum over many years as part of the council’s cultural revenue grants programme.
"All recipients were informed that that funding stream was ending to be replaced by the Culture and Creative Investment Programme.
"Applications were invited to the new Investment Programme and it has become clear that this is not the appropriate funding stream to support the Royal Cornwall Museum.
"However, their work and their offer are clearly important to many in Cornwall and beyond and we want to work with them to find an alternative way forward.”
The executive director of the Royal Cornwall Museum has said he is still in a state of bewilderment at the decision, particularly as it could impact "the bigger picture" of investment in Truro.
Jonathan Morton told Cornwall Live: "It's been a real shock. We were bewildered and surprised by what we found out last Thursday. We've been processing the news."
He said the council's decision was particularly upsetting as it had come at a time when the museum was planning on growing on its success, saying the decision will have ramifications for the museum's relationship with other funders.
"From all the amazing feedback we've received over the last 24 hours it really goes to show the role the museum plays and what an important place it is within Truro and Cornwall," he said.
"We're wanting to build on that and invest in the building and that's why we've been wanting and needing to secure additional funding, both through the local authority and further in the arts.
"So many other museums, which are comparable to the Royal Cornwall Museum, receive significant local authority funding and the Royal Cornwall Museum is quite unique in the amount it did receive and even more unique in not receiving any now. That's unusual for this sort of organisation.
"For us, it's a bit of a step beyond being the only county museum not funded by a local authority. It's not a question of us operating a museum with no local authority funding, there won't be a museum."
In the short term, a meeting is being arranged between the Council and the museum to discuss the immediate impact the drop in funding with a follow-up long term plan this summer to keep the museum open.