The Northern Irishman stepped down as Celtic boss on Thursday morning, and O'Neill tipped his former player for success in England.
He told reporters: "Why shouldn't he be [a success]? He's got really good experience now and he's had the experience of managing I believe, one of the best clubs in the world, and he has taken that on and that will stand him in good stead.
"I know him particularly well because I signed him twice as a player. Once for Leicester City and once for Celtic and he did exceptionally well in both spells."