Cole's injury was diagnosed when he experienced weakness during a gym session at St George's Park.
The Leicester Tigers tighthead has been told to rest the problem and will now undergo further assessment, although no date has yet been given on when he will resume playing.
"It's massively disappointing for Dan, Leicester and England," said England head coach Stuart Lancaster.
"But the important thing is that this has been diagnosed and that Dan gets the appropriate rest and treatment.
"Our medics are working closely with Leicester's to come up with a programme that will ensure Dan returns in the best possible shape.
"In the meantime, it's great to have David Wilson back, as well as Henry Thomas who has trained well and came off the bench against Scotland and this will be an opportunity to develop experience in this area."