O'Shea will feature in his final senior appearance for the Republic of Ireland on Saturday as they take on the USA, after having announced his retirement from international competition.
Keane played alongside the Sunderland defender both internationally and at Manchester United and recalls the 37-year-old's impressive mental strength.
He told 'Sportsmail': ''I remember him as a young player at United and thinking, 'This kid's got a chance,' decent player. Strangely enough, he always seemed really balanced.''
''We've talked about all the pitfalls in football, but John always seemed to be really switched-on. But listen, he could be a raving lunatic when he's at home in the evenings...I hope he is.''
Keane jokingly said: ''I hope he's a head-case and we're all proved wrong when he's locked up in six months and we're all saying 'Jesus, I never saw that coming. No, but I don't think so. He'll probably have a nice steady life, probably become a coach or a pundit.''