The operation was carried out as part of his treatment for Focal Segmental Glomerulosclerosis.
A Manchester United statement said: “Manchester United Ambassador Andrew Cole has undergone a kidney transplant operation at Manchester Royal Infirmary as part of his treatment for a condition called Focal Segmental Glomerulosclerosis.
“During his recuperation, Andrew will be taking a break from his role as Manchester United Ambassador.
“Andrew and his family would like to thank the club and fans for their support. They also request that he can continue with his treatment in private, and ask that their privacy is respected.”
Cole is the third-highest scorer in Premier League history with 187 goals to his name.
The 45-year-old spent six years at United between 1995 and 2001 and also enjoyed spells at Newcastle, Blackburn and Manchester City.