Espanyol are already lining up former Malaga manager Javi Gracia to replace the 52 year-old, with sources at the club admitting that a change of head coach is imminent.
Stoke held talks with Sanchez Flores this week but the ex-Watford manager said on Saturday: "Here I have all I need".
Flores had been expected to tell his current club Espanyol that he was bound for the Premier League strugglers. I do not understand that you put so much effort in coach.
Quique Sanchez Flores has confirmed that he will remain as the manager of Espanyol and won't become the new manager of Premier League side Stoke City. I am absolutely happy with my situation, I do not have to explain anything else.
"But the important thing is that we think".
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The ex-Atletico Madrid and Benfica boss added: "I'm very grateful to Espanyol, I've been frustrated sometimes because we haven't advanced as fast as desired, but that's because Espanyol haven't been able to, not because they haven't wanted to".
'We feel happy. We are in a happy place and we don't want to break that up'. I'm in a great city and am involved in one of the best leagues in the world.
Talk of O'Neill departing for Stoke was quietened once Flores's name entered the discussion.
Indeed, Keane is thought to have helped persuade O'Neill to stay on over the last few weeks, but there is a growing risk Stoke will make an offer that is too good for them to turn down.
O'Neill and Roy Keane were the initial favourites for the hot-seat at the bet365 Stadium after Mark Hughes was sacked.