Police Scotland said a further 18 people were taken to hospital, and three were seriously hurt.
The crash happened at about 13:15 on Saturday on the A76 by the Crossroads Roundabout near Kilmarnock.
One passenger said the bus, which had 37 people on board including the driver, had "swerved" before landing on its side on a grass verge.
In an updated statement Police Scotland said: "We can confirm a 39-year-old man has died following a serious coach crash on the A76 near Crossroads Roundabout.
"A report will be prepared for the Procurator Fiscal.
"Eighteen people are receiving treatment in hospital, three of whom are in a serious condition.
"This includes seven men under 20 years of age. Three men and three women aged between 30 and 59 and five men over 60. Enquiries are ongoing."
There are reports that the man who was killed was originally from Northern Ireland.
According to other reports on social media, the bus was carrying members of the Nith Valley Loyal Rangers Supporters Club.
The club posted on its Facebook page: "The club would like to thank everyone for their good wishes & concern at this sad time."
Ryan Rowe was travelling to work when he saw the bus, just minutes after the crash.
"Within two minutes the fire service, ambulance and police all turned up. There were still people trapped inside," he said.
"One man I spoke to was just sitting down and in tears and he was shaking. He said the bus has swerved.
"Another young man had cuts all over his arm and blood all the way down it. He was holding it in the air to try and stop the bleeding.
"Two or three windows on the bus had been smashed and when I was there paramedics were still trying to get people out of the other side that the bus was lying on."
His father Ricky said there was no sign that any other vehicle had been involved.
He said: "We'd seen people in tears and knew that it was a horrific accident - and just by the nature of where the bus was which was quite high up on the verge by the roundabout.
"We knew that it must have been going at quite a fast speed to actually reach up there on its side."
Rangers were playing Partick Thistle at Ibrox stadium on Saturday afternoon.
After the match, Rangers assistant manager David Weir offered his sympathies to the victims of the crash.
He said: "Obviously there are things more important than football. Our thoughts are with the people involved, the families involved, and I'm sure the club will be very supportive in regards to anything
they can do to help the situation."