Following Saturday's goalless home draw against Southampton - where captain Grant Holt saw a stoppage-time penalty saved - the Canaries are nine points clear of trouble heading into the final nine games.
However, Bunn insists there is a bit more hard work ahead before anyone at Carrow Road can start talking about a third season in the Premier League.
"We don't see ourselves as safe. I think we need a couple more wins and then we will be okay," said Bunn, who is hoping to retain his place in the side when England goalkeeper John Ruddy is back in contention later this month after recovery from his long injury lay-off following thigh surgery.
"We have got some crucial games coming up.
"Sunderland is a winnable game so hopefully we can look forward to that and get the three points, then against Wigan too.
"They are games against teams around us, so our main aim is not to lose those games, try to take a point away from home and win our home games.
"If we do that then we will be all right."
Norwich boss Chris Hughton is also not about to lose focus over the run-in.
"You are never close enough to survival until it is impossible," he said,"
"It is certainly going to be tense in these final nine games for everyone involved."
After Norwich head to Sunderland for Sunday's live match, Hughton will see a large proportion of his squad head off on international duty.
Defender Sebastien Bassong is set to play for Cameroon, while on-loan forward Kei Kamara will head off with Sierra Leone and Simeon Jackson continues to feature for Canada despite his limited playing time for Norwich this season.
Closer to home, defender Russell Martin and winger Robert Snodgrass have both been called up into the Scotland squad by Gordon Strachan for the World Cup qualifiers against Wales and Serbia.