The 74-year-old Italian took charge of the team in 2008, but a poor World Cup qualifying campaign has seen him step down as coach.
Ireland have lost their last two games in qualifying, most recently a 1-0 defeat against Austria in Vienna on Tuesday.
Trapattoni won 26 of his 63 matches in charge, drew 21 and lost 16.
He narrowly missed out taking the team to the 2010 World Cup before guiding them to Euro 2012, where they were knocked out in the group stage.
Former Aston Villa manager Martin O'Neill is the early favourite to replace Trapattoni.