The team issued a brief statement saying that Ainge, 50, was taken to a Boston hospital but provided few other details.
"Danny Ainge is recovering at Massachusetts General Hospital after having a minor heart attack. He is recovering nicely," the team said. "Mr. Ainge is expected to remain in the hospital for a few days."
A spokeswoman for Massachusetts General Hospital referred comment to the team.
The news was first reported by WCVB, a Boston TV station.
"All of Celtics Nation is thankful that Danny got incredible medical care so quickly and so effectively," Celtics owner Wyc Grousbeck said. "We wish him, Michelle and the kids all the best and know he will be back with us very soon."
A member of two Celtics championship teams in the 1980s with the original Big Three of Larry Bird, Kevin McHale and Robert Parish, Ainge was an NBA All-Star in 1988.
As general manager, Ainge helped the franchise raise its record 17th NBA championship last season by assembling a new Big Three, acquiring Kevin Garnett and Ray Allen to join Celtics veteran Paul Pierce. Ainge was named 2007-08 NBA executive of the year and in October was rewarded with a contract extension and the title of president of basketball operations.
The Celtics are scheduled to begin the playoffs on Saturday against the Chicago Bulls. Earlier Thursday, the Celtics said Garnett was unlikely to return from a knee injury to play in the postseason -- a potentially devastating blow to their hopes of a repeat championship.
Ainge missed Wednesday night's regular-season finale because he wasn't feeling well. He woke up in the morning with chest pains.
Ainge was a standout basketball player at Brigham Young, winning the Wooden Award awarded to the nation's top college player in 1981. He also played three seasons of pro baseball, batting .220 over 211 games for the Toronto Blue Jays.