HITC Sport revealed Bournemouth's interest in re-signing Grabban at the weekend – with the Canaries quick to knock back initial advances from the Cherries.
Two bids were shunned by Norwich and it is now understood that a third offer was rejected on Tuesday.
Bournemouth, though, are determined to get Grabban and are expected to return with another bid.
HITC Sport understands the player is open to the idea of moving back to the South Coast.
The 27-year-old only moved to Carrow Road last summer, and netted 12 goals in 36 appearances as he helped Norwich to secure promotion to the Premier League alongside Bournemouth.
Prior to that, he had spent two seasons with the Cherries, notching 35 times in 95 outings to catch the eye of rival outfits.