Former Celtic and Hibs boss Neil Lennon is hot favourite for the Ireland job.
Incumbent manager Stephen Kenny is under intense pressure having overseen a run of one win in 15 games since taking the job last year.
His first win only came in a friendly with Andorra in June and Wednesday's defeat to Portugal leaves them pointless after three games in their World Cup qualifying group.
Talk has begun to circle over Kenny's potential successor and a few known names are in the frame.
Roy Keane, Robbie Keane, Keith Andrews and even one-time Celtic target Eddie Howe have all been linked.
But Paddy Power have made Lennon 23/10 favourite to be named their next permanent manager.
Lennon managed Bolton Wanderers and Hibs in between his two stints as Celtic boss but is currently out of work after his second spell at Parkhead reached a messy end.
He made 40 appearances for Northern Ireland as a player but could now be in the running to take over Republic of Ireland if they decide to move on from Kenny.
The current boss will have at least two more games to save his skin, however, with Azerbaijan and Serbia both travelling to Dublin for World Cup qualifiers.
They desperately need a pair of wins to stand any chance of qualifying for the tournament
