"It's massively disappointing for Dan, Leicester and England," coach Stuart Lancaster said in a statement.
"But the important thing is that this has been diagnosed and that Dan gets the appropriate rest and treatment.
The injury was discovered when Cole experienced weakness during a gym session at St George's Park last week.
Cole, 26, has been capped 45 times by England and has developed into a cornerstone of their pack since making his debut in 2010.
"Our medics are working closely with Leicester's to come up with a programme that will ensure Dan returns in the best possible shape," Lancaster said.
"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."
England host Ireland at Twickenham on Saturday having lost their first match in France before beating Scotland last weekend.
(Reporting By Josh Reich, editing by Ed Osmond)
