Stoke's board, led by the chairman Peter Coates and his son John, are understood to be considering the credentials of Quique Flores, the Espanyol coach, but O'Neill is their preferred candidate and discussions are scheduled to continue today.
It is also believed that O'Neill has only made a verbal agreement with the Football Association of Ireland to extend his contract to the Euro 2020 campaign.
"It's like being engaged - you shouldn't be going out with other people".
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.
Stoke have had to rejig their managerial wanted list after Gary Rowett signed a new deal on Tuesday night to stay in charge at Derby until 2021 with former Watford boss Quique Sanchez Flores, now in charge of La Liga club Espanyol, emerging as the frontrunner for the Stoke job.
At least 22 die in brutal cold across US
In Hartford, Connecticut, a brutal cold of minus 12C (10F) yielded a wind chill of minus 28C (minus 20F). In Burlington, Vermont, the temperature was minus 1 on Saturday morning, with a wind chill of minus 30.
The Irish Independent does, however, speculate that, should the package at Stoke fail to find room for Keane, he will remain with the FAI and another of the Irish backroom staff - Steves Guppy, Walford or Seamus McDonagh - will join O'Neill.
It means O'Neill doesn't need permission to talk to interested parties, nor would the FAI be entitled to any compensation. The contract was agreed in principle, but O'Neill delayed.
Those comments were aired in the immediate aftermath of Ireland's hammering at the hands of Denmark in the November World Cup play-off.
RTE pundit Dunphy has been critical of O'Neill in the past but this time has taken aim at the association. It appears that O'Neill, whilst having agreed a contract with the FAI was still open to offers from others.