Lambert does not feel it is too late for Sunderland boss O'Neill - who faces his former club Villa on Saturday - to be given the opportunity to take charge of one of the elite sides.
Asked if he could see O'Neill in charge of one of the country's biggest clubs one day, Lambert said: "Definitely. I wouldn't say no to that, that's for sure. Yes, I'm surprised he's not had the chance yet because he could do that, no problem, that's for sure."
Sunderland have won only one league game so far this season and crashed out of the Capital One Cup to local rivals Middlesbrough in midweek.
Lambert said: "Martin won't be happy with not getting through the cup tie because I know what type of manager he is.
"But he will turn things around. I know what he's like, he's really competitive. He'll want to win the next game."
Lambert is full of praise for the way O'Neill revived Celtic's fortunes at a time when Rangers were the dominant force in Scottish football.
He said: "I don't think people realise how big that job is. Rangers had a right top side at that time as well.
"I think at that time when he came in, I think we'd just lost the league by 21 points or something.
"When he first came in, he told us what was going to happen. Lads started to come in and, with the ones that were already there, he just got a right good team together."