The Daily Mail reports the voicemail message to a member of QPR's coaching staff was left on Sunday at 1.30am. It mocks former England keeper Green's performance during the 5-0 opening day home defeat by Swansea.
In the message, Kenny, who appears to have been celebrating Leeds's victory over Wolves on Saturday lunchtime, says: "It's Rob Green's agent here, I just wondered, er, how you got on today. I didn't really see the f****** score but I heard it didn't go too well.
"But, never mind, Robbo's a good keeper. Don't you worry about that, he's a good keeper."
Kenny then hung up.
The content of the voicemail was disclosed by a QPR source. It has been obtained with the full consent and knowledge of the club official concerned and was one of a number of abusive messages left by the Leeds keeper that night.