Relegation-threatened Livorno have sacked Domenico Di Carlo after just 88 days in charge and re-appointed Davide Nicola.
Virgil van Dijk claims he knows nothing of interest from English clubs and is keen to prove himself in the Champions League with Celtic.
The 61-year-old Northern Irishman and number two Roy Keane each signed initial two-year deals on Tuesday evening as the Football Association of Ireland's search for a replacement for Giovanni Trapattoni finally drew to a close.
O'Neill, the FAI's top target, took his time over the decision to accept its offer of employment, but ultimately could not pass it up.
He told Sky Sports News: "It's a privilege to do it. You know, a job like this may not come round again for quite some considerable time.
"The Republic of Ireland have had some success - Jack Charlton was great, Mick McCarthy excellent, and Trapattoni did very, very well indeed, and all stayed in the job for quite a number of years, so that opportunity might not be afforded to anyone in the foreseeable future."
O'Neill and Keane will get straight down to work next week when the players meet up ahead of friendlies against Latvia and Poland.
However, the longer-term aim is qualification for the Euro 2016 finals - a process which will begin in September next year.
Ireland, of course, went to Euro 2012 in Poland and Ukraine under Trapattoni, but they parted company with the 74-year-old Italian in September after successive defeats by Sweden and Austria left their hopes of securing their passage to next summer's World Cup finals in Brazil in tatters with two games still to play.
O'Neill is ware of what is expected of himself, Keane and their players, but is relishing the task ahead.
He said: "I am sure that the players will be very positive. I think, generally speaking, they want to play for their country anyway, regardless of who is the manager.
"There will be some expectation too, but I think we can live with that."
O'Neill's decision to recruit Keane as his number two raised eyebrows when it was first mooted, not least because of the former Republic skipper's frosty relationship with the FAI and its chief executive John Delaney in the wake of his unscheduled departure from the 2002 World Cup finals squad in Saipan.
However, the former Wycombe, Norwich, Leicester, Celtic, Aston Villa and Sunderland boss is convinced that is history.
He said: "It's been a long, long time. I know that I have spoken to John Delaney - John Delaney is very, very pleased indeed that Roy is coming on board.
"Roy is naturally delighted himself and I think, going forward, everything should be fine."
The pair will head for the dugout at the Aviva Stadium for the first time next Friday evening when the Latvians visit Dublin, and will then travel to Poznan and their clash with the Poles four days later.