The current Tottenham man is set to make his loan move from Spurs a permanent one in the coming days after shining for the Tigers as they ensured a second season in the top flight and booked a spot in the Europa League by reaching the FA Cup Final.
But even with the 24-year-old on his way to the KC Stadium, Bruce wants Snodgrass to further bolster his ranks.
The Scotland international was one of the few shining stars for Norwich last season as difficult late fixtures contributed to their drop to the Championship.
Even though he is known to have a strong affection for the club, the 26-year-old is far from keen on playing outside of the big time at this stage of his career, and has already been linked with a move north or the border to join Celtic.
But Hull are ready to offer him a Premier League lifeline, and are confident that a bid of £6m will be enough with his contract now entering its final 12 months.
QPR, Sunderland and West Brom are also in the running.