O'Neill created history by leading Norn Iron to their first-ever knockout appearance at Euro2016 before their campaign was finally ended when Gareth McAuley's own goal gave Wales a 1-0 victory at the Parc des Princes on Saturday.
The Saints are manager-less following Ronald Foeman's move to Everton and Davis suggests perhaps O'Neill could already be prepared to take the reins at such a club.
"Stranger things have happened haven't they? So who knows?" the Northern Ireland skipper said when asked about the possibility of O'Neill joining the Saints.
"As a player or a manager everyone needs an opportunity. Our lads playing on this stage here are playing lower leagues and they have shown they are more than capable of getting to this standard.
"You need a chance to do that and it is the same for managers. It is somebody having the belief in Michael to give him the opportunity and I am sure he wouldn't let anyone down.
"There is going to be people looking at him. He has done a hell of a job here with us. We have had a hell of a progression in the last couple of years.
"We were unlucky not to go into the quarter-finals. I am sure there will be people thinking he is a possible candidate for another role but we would like to keep him obviously.
"I am sure at some point he will go on to manage a big club."
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