O'Neill was initially wary of Stoke's interest as they appeared to have made former Watford manager Quique Sanchez Flores their preferred target, but Telegraph Sport understands they had kept communication lines open even though the 65-year-old had told them on Friday that he was no longer interested.
Espanyol held a press conference on Saturday afternoon that supporters though would spell the end of Sanchez Flores' time at the club, but it's quite the opposite.
Flores spent the 2015-16 season in the Premier League with Watford.
Sanchez Flores, who is under contract with La Liga side Espanyol until 2019 but has a release clause in his deal, had been Stoke's first choice to replace Hughes.
"I belong to Espanyol".
Livermore escapes FA sanction after fan bust-up
Taking into account the sensitivities around the incident, the FA does not believe Livermore should be subject to any formal action.
The 52-year-old has been strongly linked with the Potters following the dismissal of Mark Hughes. "I want to continue being happy with my profession".
"I don't want to talk about me, but to enjoy these moments".
O'Neill and his coaching team agreed contract extensions with the Football Association of Ireland last October.
Stoke had hoped to have their new manager in place for the Premier League game against Manchester United on Monday, but that now looks unlikely. Clubs are interested in players and coaches, and this is a lifetime.
Defenders Erik Pieters and Bruno Martins Indi, sidelined of late by hamstring and groin injuries respectively, could be available for selection for Monday having returned to training this week.