The 72-year-old believes he, assistant Marco Tardelli and their staff completed the latest phase of their mission when they guided their side into the Euro 2012 finals last night.
Trapattoni said: "We said we did not want to put any pressure on the FAI but in the future, we will talk about this. It was important for us to have clarity.
"We are all responsible people and we were aware that had we failed with qualification, things could have gone differently. But now, there is clarity on both sides.
"I have said I think we deserve a new agreement, Marco and myself. We are now happy with the squad. We find new players in almost every game.
"We have done what we were asked to do by the FAI and the Irish people. They chose us for this reason and I think we have given the perfect answer with qualification."
John Delaney, chief executive of the Football Association of Ireland, indicated last night that the board would meet within the next couple of weeks and talks with Trapattoni would follow.
The Italian and his staff are part-funded by businessman Denis O'Brien, and have already taken two pay cuts with Ireland's financial difficulties biting deep.
There has been speculation that a significantly reduced package could be on the table.
Trapattoni himself has remained, in public at least, relaxed about his contractual situation, preferring instead to concentrate on the two-legged play-off tie against Estonia which secured their passage to Poland and the Ukraine.
However, he has insisted if the worst came to the worst and he could not agree a new deal with Ireland, he would not retire from the game, and having further enhanced his curriculum vitae by ending the Republic qualification drought, he would certainly be in demand.