The 64-year-old - currently a first-team coach at Celtic Park - experienced what Celtic describe as a minor attack on Thursday.
McGrain is "recovering well" and is expected to leave hospital on Friday.
A Celtic statement read: "Clearly, the thoughts and best wishes of everyone at the club are very much with Danny as we wish him a full and speedy recovery."
McGrain, a full-back in his playing days, won seven league winners' medals in 20 years at Parkhead before leaving in 1987.
He played 62 times for Scotland, featuring in two World Cup finals tournaments.