Cole, who has scored more Premier League goals than anyone other than Alan Shearer and Wayne Rooney, has been forced to withdraw from a charity match at Old Trafford this month.
And in a statement released on the United website the club ambassador said he had been diagnosed with a condition called Focal Segmental Glomerulosclerosis in June after contracting an airborne virus.
Cole said: "I spent close to three weeks in hospital under expert medical care, undergoing a kidney biopsy and temporary haemodialysis (dialysis).
"During this time I have been unable to work or to carry out my normal day to day activities and I am under close supervision from the renal team at the Manchester Royal Infirmary.
"The good news is that I am feeling better as each day passes and the consultant is hopeful that I will make a full recovery by next spring.
"The messages of support from the fans have been gratefully received."
Cole won five Premier League titles, two FA Cups and a Champions League after joining United from Newcastle in 1995.