The Great Central Railway's plan for a new museum, to be built in Leicester, is expected to cost £15million in total.
Alongside the proposed £10million lottery grant, submitted to the National Heritage Fund, £5million of additional match funding would be required.
A striking building is planned for construction, alongside the Great Central Railway's Leicester North station where the museum will be connected to the running lines, allowing for inter-changeable displays.
Drumming up support for the project, City Mayor Sir Peter Soulsby said:
This is a really exciting opportunity to create a heritage railway centre of national standing in Leicester...
...The museum will house items which are of both national importance and local interest, complementing the nearby National Space Centre and Abbey Pumping Station, helping to enhance the city's reputation as an important visitor destination.
More top news
A major inquiry has been launched in to the suspected abduction of a woman who is thought to have been forced in to a car against her will.
A family pet was repeatedly kicked to within an inch of its life by burglars who broke into a home, police have said.
Thousands of fans - who could not fit inside the cathedral - listened to proceedings from outside.