Lennon brought the curtain down on his Celtic career yesterday just 11 days after celebrating a third successive Premiership title. The likes of Henrik Larsson, Malky Mackay and Davie Moyes have all been linked with the Hoops top job so far.
But former boss Hay reckons the Parkhead hierarchy should turn to Keane to fill the managerial hotseat.
He told SportTimes: "I have mentioned him as a possible Celtic manager before. He was a top class player, he had a spell at Celtic and knows what the club is about.
"He has managerial experience at Sunderland and Ipswich and done well at both clubs and is now with Martin O'Neill at the Republic of Ireland.
"He is the type of person I think would do a good job at Celtic. I know he could be seen as a controversial, but I think it would be a good move for Celtic."
Lennon's decision to step down and bring an end to his four-year tenure in the Parkhead dugout was met with shock yesterday, but thoughts will now turn to the future and who will replace him as boss.
The champions will be overwhelming favourites to claim the title once again next term, but it is in the European arena where the new man will have to shine.
Lennon lead his side to successive appearances in the Champions League group stages and Hay knows it is a huge summer for the Hoops.
He said: "Whoever the new manager is, they are going to be taking over a fantastic club. They are in a great financial position, they will win the Premiership title again next season and they have the Champions League to look forward to as well."