O'Neill - who held talks with Stoke over the vacant manager's position - is flying to Dublin along with his backroom team for tonight's Soccer Writers' Association of Ireland awards as it now appears he is out of the running.
"If he gets a big job, it's still a big job, a Premier League job, Stoke coming in for him the way that they are, you think he's probably the sort of manager that could go in there and get that short-term for them".
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.More news: Rob Lowe schools Bella Thorne after insensitive traffic tweet
Given the requirement to regain stability at the Bet365 Stadium following a disastrous 2017, sources claim Stoke would want O'Neill on an undivided basis.
Dunphy also called on the FAI to explain why a new contract had not been signed after the deal was verbally agreed in October.
That is despite the fact he is still operating as Ireland manager.More news: USA said to fatally bomb Afghan militia
Should Keane not leave then Steve Walford, Steve Guppy and Seamus McDonagh are in contention for roles.
"I can't think of any other Association that would allow their manager to stall in this way", Dunphy told News At One on RTÉ Radio 1.
Those comments were aired in the immediate aftermath of Ireland's hammering at the hands of Denmark in the November World Cup play-off.More news: United Kingdom minister resigns after rejecting PM's reshuffle
The FAI insisted that yesterday was no different to any other day, and that O'Neill would be flying to the UEFA Nations League draw later this month as Irish manager.