It is believed Blues offered a £3million down payment and a further £2.5million based on their Premiership survival over the next two seasons and Rovers want rejected that offer.
Now it is understood Everton boss Moyes is ready to join the chase for the talented 23-year-old midfielder with a possible player-plus cash offer to Rovers.
Dunn has rejected a new deal at Ewood Park although he has said that just because he has turned down Rovers first offer of an improved deal that does not neccessarily mean he wants to leave the Lancashire club he joined as a youth.
Meanwhile newspapers are reporting that Bruce is also keen on Arsenal striker Francis Jeffers and Manchester United goalkeeper Roy Carroll as he looks to maintain Blues' stay in the top-flight.