Speaking shortly before the transfer was confirmed, Spurs boss Harry Redknapp told Sky Sports News: "It's a good move. Robbie needs to play.
"He's too good a player not to be starting here regularly. He wants to play all the time.
"It's been difficult to guarantee Robbie a starting place at the moment.
"He's not a happy man when he's not playing regularly.
"But he loves his football; he's a great trainer, a great man about the place."
Redknapp believes Keane will relish the move, saying: "To go to Celtic is a dream come true for him.
"He's a big Celtic man and he'd like to go up to Celtic and play like Roy Keane did.
"The Irish lads, it's in their blood and they dream of playing for Celtic one day.
"This opportunity is one they want to take and we don't want to stand in his way, because we're strong in that position."
Keane, who has been linked with moves to the Hoops in the past, was on Monday travelling to Glasgow and was set to be unveiled at an 11pm press conference at Celtic Park.
The forward will be the club's fourth signing of transfer deadline day, topping those of Diomansy Kamara, Edson Braafheid and Paul Slane.
The day also witnessed the ninth departure from the club since the turn of the year, with Scott McDonald joining Middlesbrough for an undisclosed fee.
Keane signed for Spurs in 2002, making almost 200 appearances and scoring 80 goals, before joining Liverpool for around £20million in July 2008.
However, he went on to make only 15 appearances for the Anfield side and rejoined Spurs in February 2009 for £12million, taking his cumulative transfer fees to more than £70million.
His previous clubs include Wolves, Coventry, Inter Milan and Leeds.
Celtic chief executive Peter Lawwell said: "We've been pretty persistent in the last few windows in terms of keeping in touch with what he was wanting to do.
"Thankfully, today, we were able to do a deal.
"We've known for a while that Robbie is a Celtic supporter and he's a great connection with the club and our supporters.
"Up until today, things were not looking very great at all. Things changed this afternoon and were able to put a deal in very quickly.
"His situation at Spurs was made, I think, pretty plain to him. He always knew we were very, very keen to get him."
Lawwell said there was no specific option to make the move permanent but did not rule out doing just that in the summer.
He added: "What we find is that players that come to Celtic very rarely want to leave."
Lawwell refused to reveal how Keane's salary - reportedly in the region of £65,000 a week - would be paid and denied his signing was merely a ploy to appease disgruntled supporters.
"We're in a transitional period at the moment," he said.
"Tony probably wasn't blessed with the strongest squad we've had here when he arrived.
"We were always looking to bring in quality, our supporters do deserve the best and that's what we do at Celtic."
Kilmarnock v Celtic 1st Goalscorer: Keane, R. 3/1