Boro have become the first club to make an official move for Grabban, who was one of the Championship's top scorers last term.
The 26-year-old was close to joining Brighton in January when they met a £1.1million release clause only for Bournemouth to persuade him to sign a new deal.
However, Sky Sports revealed earlier this month that Grabban's new deal had a £3million release clause and a number of clubs look ready to meet that mark.
The first team, though, to make an approach are Boro and the Teesside club are hoping to be able to seal a quick deal.
But Boro will also be aware that rival clubs are waiting in the wings, with recently relegated Cardiff City set to make an offer in the coming days.
Crystal Palace and previous suitors Brighton are also monitoring the situation.
Boro, though, are hoping they can persuade Grabban to make the move to the north east, having made him their number one summer target.
With Marvin Emnes and Lukas Jutkiewicz likely to move on, and with Danny Graham returning to parent club Sunderland, manager Aitor Karanka is looking to heavily invest in his attacking options and will hope Grabban is the man to lead his line going into next season.