Chris Hughton's deputy netminder shone in Sunday's win over Sunderland in the absence of the injured John Ruddy, and wants to catch the attention of Trapattoni if he gets an extended run in the team.
"It is one of my aims to try and break into the squad and if the opportunity arose I would definitely be interested in playing for them," said Bunn.
"I think every player wants to play international football and play at the highest level and I am no different.
"I believe the goalkeeping spot with the Republic is up for grabs since Shay Given retired. Keiren Westwood is first-choice at the minute, but he is not playing regularly for Sunderland.
"I now have a run of games in the Norwich team and it is a good time for me to show what I can do and impress."
As it stands, Trapattoni is more concerned with Bunn's club colleague Anthony Pilkington, who has spoken with FAI staff about his situation without fully throwing in his lot behind the Irish cause.
The goalscoring winger was capped by Ireland at U-21 level, but could still play for his native England.
Meanwhile, former Irish international Caleb Folan has signed up for another new adventure after joining the wonderfully named Malaysian Super League side T-Team.