Clive Wright, the authority's chief executive, said the option is one of a number being explored by the council as it seeks to conclude an agreement over the management and funding of its museums by September.
Other options open to the council are the creation of a museums trust or individual town and parish councils stepping in to help.
Museums receiving funding from Shropshire Council include Acton Scott Historic Working Farm, Coleham Pumping Station, Ludlow Museum, Ludlow Museum Resource Centre, Much Wenlock Museum and Shrewsbury Museum & Art Gallery.
The authority has said that it intends to cut its £718,000 museums' budget to zero as part of cost cutting measures.
Mr Wright said: "There are some options on the museums. We could set up a trust. There are other partnerships being discussed. One is Ironbridge Gorge Museum Trust, among others, about how we can run museums.
"We are also in discussions with local town and parish councils about how they can run in future.
"We have had interest from a private company and the proposition there would be to take the viable museums and run them as an entity.
"But they would be looking not just at Shropshire but other localities as well for economies of scale."
Mr Wright said the council had not yet sought to put any requirements on limits for charging for entry to the museums affected.
He said that would be a subject for more detailed discussions as the issue progresses.
Anna Brennand, chief executive of Ironbridge Gorge Museum Trust, confirmed the organisation would be meeting with the council for discussions.
She said: "The trust is due to meet with Shropshire Council in the foreseeable future to discuss how we can work more closely and in partnership with the council."
Earlier this month, Lottie James, chairman of the Friends of Ludlow Museum, said the council should look to establish a trust to manage the future of the affected museums.
She said: "The friends are very keen that the Shropshire museums service goes into a trust. It's time Shropshire Council gave this fresh consideration.
"What they want to do is dump them on to town councils, and town councils do not have the money and are just not up to it.
"Shropshire Council is cutting it fine. They've got less than a year now until the budget is cut, and they're telling us that if we want to do something we've only got five months."