Bunn is thought to be on Giovanni Trapattoni's radar as the Ireland coach looks to bring in new players to his squad.
The 28-year-old was born in Southgate, London, but qualifies to play for the Republic through his father and grandmother.
The goalkeeping spot in the Republic squad is up for grabs following the retirement of Shay Given after Euro 2012.
Sunderland goalkeeper Keiren Westwood is the current No.1, but Westwood is not a regular starter for his club with Simon Mignolet the first-choice in Martin O'Neill's side.
Bunn has been outstanding for Norwich this season and Canaries and England goalkeeping coach Dave Watson is thought to be a huge admirer of the former Blackburn shot-stopper.
Bunn, who was named man of the match in Sunday's win over Sunderland, admits he would jump at the chance of call-up to the Republic squad.
"It is one of my aims to try and break into the Republic of Ireland squad and if the opportunity arose I would definitely be interested in playing for them," Bunn told Sky Sports.
"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 really 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 hopefully."