The Norfolk club want to add the 23-year-old Republic of Ireland wing-back to their squad after winning promotion back to the Barclays Premier League in the Sky Bet Championship play-off final at Wembley.
Bruce, though, is not about to let the Canaries land one of Hull’s key men on the cheap, despite their own relegation from the top flight.
“Norwich have come in with bids for Robbie Brady, but I have to say they have been embarrassing,” Bruce said in the Hull Daily Mail.
“The supporters should know the truth and Norwich came in two weeks ago with a bid of £2m, £2.5m and then £3m.
“That’s ridiculous for someone who is 23, has played two years in the Premier League and has become a regular international.
“When we have just got almost twice that for Tom Ince (who joined Derby for £5m). It doesn’t make sense. We paid £2.75m for Robbie and that was three years ago.
“This can unsettle players, do this, that or the other, and it is all part of the game, but Robbie Brady will not be going for anywhere near that.
“If it happens, it happens, but Norwich are nowhere near the figures we would even think about accepting at the moment.”
Norwich have so far added only midfielders Youssouf Mulumbu and Graham Dorrans, whose loan move was made permanent, to their squad for next season.
Club-record signing Ricky van Wolfswinkel could also be back in contention after reports in Portugal suggested a proposed move back to former club Sporting Lisbon fell through for the Holland forward, who spent last season on loan at St Etienne.
Mulumbu moved to Carrow Road on a free transfer after his contract expired at West Brom, where he had spent six and a half seasons.
The 28-year-old Congo international is confident he can help Alex Neil’s Canaries hold their own back in the top flight.
Mulumbu said to the club’s official website: “I am really looking forward to getting back to what I do, and to be playing at Carrow Road. I am 100 per cent confident it is going to be a great season for myself and Norwich.
“We have had two weeks training together and I have seen the quality we have.
“I had six or seven years at West Brom, but now I have a new challenge and new ambitions to win my position here and keep it. That’s going to drive me forward and make me step up my game in order to improve again.”