Cole, who won the treble with United in 1998-99 and scored 121 goals for the club in 275 appearances, had the transplant as part of his treatment for the nephrotic syndrome focal segmental glomerulosclerosis.
"Andrew and his family would like to thank the club and fans for their support," United said on their website.
"They also request that he can continue with his treatment in private, and ask that their privacy is respected."
The 45-year-old Cole joined United from Newcastle United midway through the 1994-95 season and scored five goals in a 9-0 rout of Ipswich Town. The following season he helped United win the title, pipping Newcastle to the crown.
He later struck up a prolific partnership with Dwight Yorke.
After leaving United Cole had spells with Blackburn Rovers, Fulham and Manchester City, taking his Premier League goal tally to 187, behind only Alan Shearer and Wayne Rooney.
(Reporting by Martyn Herman)
