The 61-year-old has received a series of approaches from the Football Association of Ireland and Press Association Sport understands he is tempted by the opportunity to resume his career on the international stage.
O'Neill has been out of work since being sacked by Sunderland in March and has since been linked with a series of Barclays Premier League vacancies, both current and predicted, in recent weeks.
However, the FAI identified the Ulsterman as its number one target within days of Trapattoni's departure in September and while it has drawn up contingency plans in case it is unable to strike a deal with him, it is understood the governing body is increasingly optimistic that he will respond positively to its overtures.
No firm agreement is yet in place, but with the association pushing for a decision sooner rather than later, significant progress could be made within days.
Former Ireland international Ray Houghton and high performance director Ruud Dokter were asked to assess the field and draw up a short-list of potential candidates following 74-year-old Trapattoni's exit in the wake of successive World Cup qualifier defeats by Sweden and Austria.
Mick McCarthy, who took Ireland to the 2002 World Cup finals in the Far East, and his captain - until their infamous fall-out in Saipan at least - Roy Keane have also been heavily linked with the job.
Intriguingly, the possibility of Keane serving as a number two to O'Neill has also been mooted in Ireland, although he is a man who still divides opinion in his native country.
Businessman Denis O'Brien, who part-funded Trapattoni's five and a half-year reign, has agreed to continue his financial support with the FAI looking to make an eye-catching appointment.
It did just that when it persuaded the much-travelled Trapattoni to take up the challenge of restoring the nation's football fortunes back in 2008.
He guided the Republic to the play-offs for the 2010 World Cup and into the Euro 2012 finals via a similar route.
However, a poor showing in Poland and Ukraine heralded a decline which saw the team finish in a distant fourth place in Group C behind Germany, Sweden and Austria as they attempted to reach next summer's World Cup finals in Brazil, and that proved fatal to the Italian's tenure.
Under-21s boss Noel King took charge for the final two qualifiers against Germany and Kazakhstan last month.
Ireland face Latvia in Dublin on November 15 and Poland in Poznan four days later in friendlies, and it remains to be see whether the new man will be in charge for those fixtures.