The 74-year-old had planned talks with the board following three consecutive defeats, including losses to Rangers either side of a Europa League setback against Ferencvaros.
However, O'Neill has now confirmed his determination to continue and guide Celtic forward in their pursuit of the Scottish league title.
"I am. I'm extremely prepared for it. We had a difficult week. It can happen, you can lose to your arch-rivals, you can lose a European game. It's certainly not the end of the world, as it seemed to be described. But (I'm) ready to go," he told the club website.
"I did think I would take a little bit of time and it didn't take me that much time to think about it.
“I want to lead the club, I want to try and win the league, which is what we set out to do, and just as determined now as ever before.
“ It was a time of reflection for a moment or two and, yes, some things that you think about. But overall, I'm back, I'm raring to go and the team's ready."
Celtic return to action against Motherwell at Fir Park on Sunday.