The 64-year-old former Scotland full-back has been a member of the Hoops' backroom staff since 1997 and was promoted to first-team coach by manager Neil Lennon in October 2012.
In a statement released on their website, the club said: "Celtic Football Club confirmed today that club legend and current first-team coach Danny McGrain has experienced a minor heart attack.
"Danny is recovering well and is scheduled to be released from hospital tomorrow (Friday). 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 won eight Scottish league titles with the Hoops between 1971 and 1986 and was capped 62 times by Scotland.