It has been reported the Manchester United legend is keen on the vacant Parkhead post.
His odds have tumbled as a result and is third favourite with the bookies behind former Sheffield United manager Chris Wilder and current interim boss John Kennedy.
Keane, who played with Celtic during the 2005/06 campaign, was most recently Nottingham Forest assistant boss under Martin O’Neill.
His last managerial post was Ipswich Town ten years ago.
The 49-year-old is a Sky Sports pundit and has developed a bond with Richards.
The former defender responded to a story about Keane going to Celtic, joking he would be the assistant.
He tweeted: “I’ve just packed my bags!!! Scotland here we come.”
Former Rangers midfielder Charlie Adam issued caution for Richards.
He said: “Stay in the studio big man. There’s a change of the guard in Scotland now.”