The club announced on Friday the 45-year-old has had surgery at the Manchester Royal Infirmary as part of his treatment for a condition called Focal Segmental Glomerulosclerosis.
United added that Cole will take a break from his role as a club ambassador as he recovers.
"Andrew and his family would like to thank the club and fans for their support," United said in a statement. "They also request that he can continue with his treatment in private, and ask that their privacy is respected."
He suffered kidney failure and was later diagnosed with his current illness in 2014, after initially contracting an airborne virus.