The Potters are looking to bring in a replacement for Mark Hughes after he was axed following a dismal Premier League run and subsequent FA Cup exit at the hands of League Two Coventry City last Saturday.
O'Neill was one of several names to be initially linked with the position, while current Derby County boss Gary Rowett was also in the frame.
Rowett has since penned a new four-year deal to pledge his future to the Rams and reports claim the Potters hierarchy are keen to bring in the experienced O'Neill.
The 65-year-old has managed Leicester, Aston Villa and Sunderland in the Premier League, while he also enjoyed a successful spell as Celtic manager.
O'Neill has worked alongside Keane in the Ireland job since 2013 and the pair helped the nation reach Euro 2016, while just missing out on qualification to this summer's World Cup after losing a two-legged play-off against Denmark in November.
He has verbally agreed to stay on in Dublin, but with no deal signed it is looking increasingly likely that he could be tempted by a return to full-time club management and former Manchester United and Ireland captain Keane is almost certain to follow him to the bet365 Stadium should he agree on a move.
Former Watford manager Quique Sanchez Flores has also been linked this week but the Potters board will have to negotiate a fee with Spanish side Espanyol given that he is still under contract with them until summer 2019.