The former Northern Ireland international, out of work since leaving Sunderland earlier this year, was quickly installed as favourite for the job following Trapattoni's departure on Wednesday.
Football Association of Ireland chief executive John Delaney would not put a timescale on his search for a successor when interviewed by Sky Sports News on Thursday.
But a deal could now be done as early as next week as O'Neill is understood to be interested in the post.
Mick McCarthy and Roy Keane have also been mentioned as potential candidates, while Chris Hughton and Brian McDermott both ruled themselves out of the running on Thursday.
Trapattoni and assistant Marco Tardelli lost their jobs following back-to-back defeats by Sweden and Austria which left Ireland's World Cup hopes all but dead.
O'Neill, who represented his country 64 times as a midfielder, has been out of work since leaving the Black Cats in March but is the bookmakers' favourite to replace Giovanni Trapattoni after the Italian was sacked on Wednesday.
But O'Neill gave the link short shrift on Thursday evening, saying: "I have had no contact whatsoever from anyone at the Irish FA at this moment and there's not much more I can say about it."
It is reported that the Ulsterman would be open to an approach, while former Ireland manager McCarthy, currently in charge at Ipswich, has not distanced himself from speculation linking him to the vacancy.
Whoever the new Republic of Ireland manager is, he could face a baptism of fire in Germany.
Delaney has admitted that, in an ideal world, he would like to have Trapattoni's replacement in place in time for next month's World Cup qualifiers against the Group C leaders - who won 6-1 at the Aviva Stadium in October last year - and Kazakhstan.
Ireland's hopes of reaching the World Cup finals in Brazil are virtually over with a six-point gap behind second-placed Sweden with two games remaining.
Aston Villa manager Paul Lambert, meanwhile, believes O'Neill is an ideal candidate.
He said: "There's no doubt about it, he could do the job standing on his head. No problem.
"I've always said he's a fantastic manager and it would be up to him whether he'd want to go into international football or go back to club football. I don't know but whoever takes him has got a great manager."