Trapattoni, 72, believes he and his staff completed the latest phase of their mission when they guided their side into the Euro 2012 finals.
"We said we did not want to put pressure on the [Football Association of Ireland] but in the future we will talk about this," he said.
"I have said I think we deserve a new agreement, Marco Tardelli and myself."
He added: "We have done what we were asked to do.
"They chose us for this reason and I think we have given the perfect answer with qualification.
"We are all responsible people and we were aware that had we failed with qualification, things could have gone differently.Robbie Keane and John O'Shea are thrilled by the Republic of Ireland's qualification for Euro 2012
"We are now happy with the squad. We find new players in almost every game."
John Delaney, chief executive of the FAI, indicated after Tuesday's 1-1 draw against Estonia that his 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 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.
Having further enhanced his curriculum vitae by ending the Republic qualification drought, Trapattoni would surely be in demand.