The 49-year-old called time on his second managerial spell at Parkhead after resigning on Wednesday morning.
Celtic are 18 points behind bitter rivals Rangers in the Scottish Premiership table. His last match in charge came in a shock 1-0 defeat to Ross County.
Lennon’s assistant John Kennedy has been placed in interim charge until a successor is found.
Among the early frontrunners – according to the bookmakers – are Steve Clarke, Eddie Howe, Frank Lampard, Roberto Martinez and Rafa Benitez.
However, Sinclair believes Celtic should consider appointing former Northern Ireland boss O’Neill, who is currently a 40/1 shot with Sky Bet.
Sinclar told talkSPORT: “They’ve got to bring in an experienced manager that needs to know about recruitment, he’s got to do the scouting again.
“I think with the academy, letting players like (John) McGinn and Andy Robertson go down to England – they should be taking all the Scottish players and making them into their players.
“I like Michael O’Neill because I feel he did a fantastic job with Northern Ireland, he’s done a superb job with Stoke City and I think a manager like that – a David Moyes kind of manager who knows the game inside out, knows how to run clubs from top to bottom.
“I feel you need a manager like that to come in and get the club, give it a shake and really get it going again. I honestly feel that’s what’s needed.”
O’Neill, whose Stoke side sit 9th in the EFL Championship table, turned down the opportunity to manage the Scotland national team in 2018.
Speaking at the time, former SFA chief executive Stewart Regan said O'Neill was identified by the sub-committee on the Scottish FA as an "obvious candidate for consideration".
He said: "We requested permission to speak to Michael from the Irish Football Association, which was accepted. Michael and his advisers met our delegation in Edinburgh on Thursday and discussed all aspects of the role during a three-hour meeting.
"While this was a positive and cordial discussion, we respect Michael's decision to remain loyal to Northern Ireland, with whom he has enjoyed great success.”