The county council decided in January to replace the Snibston Discovery Museum with a smaller attraction to save money.
A plan by the Friends of Snibston to run the current museum as a charitable trust was dismissed by the authority.
The council said it followed proper processes.
The group intends to take legal action on grounds including the council's consultation on the museum's future being "flawed".
Brian Vollar, chair of Friends of Snibston, said: "We think we have a strong case, else we wouldn't have proceeded along with this plan.
"We have given plenty of indication of our intention to follow this plan and plenty of opportunity for the council to be more accommodating in following the people's wishes and not going ahead with the decision to close."
He said they would be informed in six weeks whether permission will be given for a review.
A spokesman for Leicestershire County Council said: "The council maintains that proper processes were followed to arrive at the conclusions reached by the cabinet.
"We will study the application in detail and assess the implications for the council."
Snibston Discovery Museum is focused on industry and design but also has a theatre and large country park.
Campaigners fighting to save a Leicestershire museum from closure have said they are applying to the High Court for a judicial review.