The collapse occurred in the dome part of the original building that was completed in 1994, the Bowling Green Daily News reported.
Wendell Strode, the museum's executive director, said the sinkhole appeared to be about 20 feet deep.
"As best as I can see it is about as big as the pedestal area where we had the cars displayed," he said, adding that he had not been allowed into that part of the museum.
Strode said he was awaiting an evaluation by a structural engineer before he could determine if the vehicles could be salvaged.
Authorities said they learned of the sinkhole at 5:38 a.m. when the collapse triggered an alarm that notified the fire department.