Ireland are without a win in competitive matches in 2018, and have gone six and a half hours of football without a goal.
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Mick McCarthy has been mentioned as a possible successor, while Carlos Quieroz has also declared an interest in the post.
But who else has been mentioned in conjunction with the vacancy?
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Brendan Rodgers, Celtic
Celtic manager Brendan Rodgers is often touted for other jobs due to his success with the Hoops.
Back-to-back trebles have seen him linked with just about every major managerial vacancy, and the Northern Irishman has reportedly been a target for Arsenal, Southampton, Manchester United, Spartak Moscow and Aston Villa.
Rodgers also confirmed an approach from China, which he turned down during the summer.
However, he has already insisted he has his “dream job”, despite fiery comments in early August suggesting he would quit the club if he couldn’t make the progress he desired.
Despite his achievements, the shortest odds for Rodgers are 40/1, with the likes of Betfair and Paddy Power pricing him at 100/1.
Neil Lennon, Hibs
Neil Lennon is as short as 14/1 with some bookies but is mostly priced around the 20/1 mark.
Much of the speculation linking Lennon with the post seems to stem from comments he made in the aftermath of the tempestuous Edinburgh derby last month, during which he was struck by a coin thrown from the crowd.
The 47-year-old took Hibs to within a whisker of second place last season, and in this year’s Europa League qualifiers, the Easter Road side defeated NSI Runavik of the Faroe Islands 12-5 on aggregate before seeing off Greek side Asteras Tripolis 4-3 on aggregate after coming back from two goals down at home to win 3-2. Hibs maintained an unbeaten home record with a 0-0 draw against Molde before losing 3-0 in Norway and exiting the competition.
However, when he was loosely linked with the Scotland job following the departure of Gordon Strachan, Lennon admitted he would miss the day-to-day tasks of club management if he took an international job.
Stephen Kenny, Dundalk
An attractive choice with many Irish fans, Stephen Kenny led Dundalk to within a win of reaching the Champions League group stages, and gained four points from a tough Europa League group with AZ Alkmaar, Maccabi Tel Aviv and Zenit St Petersburg. At Oriel Park he has won four league titles, two FAI Cups and two League of Ireland Cups. In six years. A brief spell in Scotland with Dunfermline ended with relegation, but Kenny also took the Pars to the Scottish Cup final, defeating Rangers, Hearts and Hibs en route to Hampden, where they were narrowly beaten by Celtic.
He would presumably be relatively cheap, which the FAI would welcome given the Association is now without the financial backing of businessman Denis O’Brien, who has spent a decade helping to fund the wages opf the Irish national team boss. Kenny is the favourite for the job with most bookmakers.
Who else has been mentioned?
Mick McCarthy is keen on a second spell in charge. The former Celtic defender is currently without a job after leaving Ipswich Town. Ex-England boss Sven Goran Eriksson, who is currently managing the Philippines national team, is among the favourites along with Steve Bruce, Sam Allardyce, Brian Kerr and Steve McClaren.
One left-field name is Jim McGuinness. The former Donegal GAA manager led the Tír Conaill Men to the All-Ireland title in 2012 and three Ulster Senior Football Championships between 2011 and 2014, and has worked with both Celtic and Beijing Sinobo Guoan in various capacities.