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 to Sweden and Austria which left Ireland's World Cup hopes all but dead.
Varicoseregent - I am well versed in Irish History so I remember learning about the 15 shot earlier that day. That was Irish people fighting for independance. The army retaliated by shooting into a sports crowd enjoying a game that they werent allowed to watch. Not really the same thing, like you said it could go on all day about history and this isnt the forum, was just giving my opinion re. the whole British/Irish thing.- new1fowler_9