HITC Sport understands that the Cherries are looking to land their former striker.
HITC Sport revealed earlier this week that Bournemouth were again looking at their former striker Grabban and were ready to make a move.
Now as they look to add at least one forward to their squad before the transfer window shuts early next month, they have made a firm offer for Grabban.
HITC Sport understands that Norwich have rejected the offer, but Bournemouth are ready to return for the 27-year-old as manager Eddie Howe is a huge fan of the former Crystal Palace and Millwall front-man.
Howe signed Grabban in 2012 for Bournemouth and he enjoyed two hugely successful seasons on the South Coast with 35 league goals, but he was snapped up by Norwich in 2014 when they met his release clause.
Now just a year later, Bournemouth want Grabban back to help their Premier League survival bid but it remains to be seen whether they can convince Norwich to sell.
Bournemouth have also been in talks with Crystal Palace to sign Glenn Murray in recent days.