The Glasgow giants will be the former Liverpool striker's seventh club of a well travelled career.
Keane's position at the club had been in doubt following the arrival of Eidur Gudjohnsen last week and manager Harry Redknapp's decision to keep Roman Pavlyuchenko.
The decision would be seen as a massive coup for Celtic.
Redknapp believes Keane needs to play first-team football he cannot offer him.
'Robbie wants to play,' Redknapp said 'and I have (Peter) Crouch, (Jermain) Defoe and Pavlyuchenko and I signed Gudjohnsen so it will be difficult to guarantee him a starting place. He has to be playing regularly so I have decided to let him go on loan until the end of the season.'
The transfer window for Scottish clubs closed at midnight, seven hours after the English window.
One late deal completed just before the English deadline of 5pm was Manchester City's capture of Adam Johnson from Middlesbrough.
The highly-rated England Under-21 international has signed a four-year deal after the clubs agreed an undisclosed fee.
West Ham bolstered their striking options by completing the signing of Benni McCarthy on a two-and-a-half-year deal from Blackburn.