The 72-year-old Italian ended Ireland's 10-year wait to reach a major tournament when he guided them to the Euro 2012 finals after Tuesday's 1-1 draw with Estonia in the second leg of their play-off in Dublin put them through 5-1 on aggregate.
Trapattoni's contract is due to expire at the end of the European Championships campaign but he says he would like to lead Ireland's bid to reach the World Cup fianls in Brazil.
He previously managed his native Italy in the 2002 World Cup in Japan and South Korea and Euro 2004 in Portugal.
Delaney told Sky Sports on Wednesday he wanted To focus on celebrating reaching Euro 2012 for now but also said there is a willingness to sit down and discuss Trapattoni's future.
"To work with Giovanni Trapattoni, for me, has been a pleasure," he said.
"The board will meet in the next couple of weeks to sit down with Giovanni to discuss... what can happen in the future. I think that should happen sooner rather than later.
"We'll leave for another week or two the debate about the manager's contract, but what I will say is: Has Giovanni Trapattoni taken us forward? Yes, he has.
"Four years ago where were we? Today we're in a far better place and we're qualified."
Trapattoni told reporters on Wednesday he believed Ireland would give a good account of themselves at the European Champioships in Poland and Ukraine next year.
"We believe, because we have confronted France, Italy, Croatia, Russia... most of the important European teams," he said.
"I have no answers but we have confronted them and I understand not only our character but the way the team is growing, our entity, our personality.
"We are happy with the team, we have trust, we have a good team and believe this team can do well."
Looking back at the campaign, Trapattoni praised the mental strength of his side following their painful defeat by France in the play-offs for the 2010 World Cup finals in South Africa.
"I am very, very proud for many reasons. We came here with an important objective," the Italian added.
"We have started the last [World Cup] campaign in a difficult group with Italy and then we had France, and we could have achieved qualification.
"But we started again and we have discovered new important options in terms of players. They have shown how important they have been for us.
"We have improved our quality, like against Russia [who they held 0-0 away]. We have started again with new mentality and the players believe in our system."
As expected Ireland were placed in the lowest of the four seeded pots for Euro 2012 by European governing body UEFA on Wednesday along with Denmark, France and Czech Republic.
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