Cole needs a period of rest after the injury was diagnosed when the Leicester tighthead experienced weakness during a gym session at St George's Park last week.
England head coach Stuart Lancaster said: "It's massively disappointing for Dan, Leicester and England but the important thing is that this has been diagnosed and that Dan gets the appropriate rest and treatment."
Cole's injury will be assessed further on Tuesday evening but it is a major blow to England's Six Nations hopes.
Sale Sharks' Henry Thomas and Bath's David Wilson are contenders to replace the 26-year-old - but Thomas' international action has been limited while Wilson has only just returned from a calf injury and as a result is unlikely to be ready to face Ireland on Saturday.
Cole has impressed for England since making his debut as a replacement for Wilson in the 30-17 win over Wales during the 2010 Six Nations campaign.
Lancaster added: "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. This will be an opportunity to develop experience in this area."
Leicester director of rugby Richard Cockerill said: "It's very disappointing for us and for England - and obviously for Dan - but his long-term fitness is more important.
"He'll have time to get over this issue and come back when the medical teams are satisfied with his progress."