Keane has a year left to run on his deal at Parkhead, where he moved after dramatically falling out with Sir Alex Ferguson at Manchester United, and will make no decision over his future until he returns from a family holiday.
According to reports in Nottingham, Burton Albion's Nigel Clough has been the long-standing favourite to take over.
Clough, who played for Forest and Liverpool as a striker, would follow in the footsteps of his father Brian - who famously lead them to two European Cup triumphs.