The London club lined up a move for former Victory Shield skipper Grimmer after he turned down a new deal at Pittodrie.
Dons boss Craig Brown was keen to keep Grimmer at the club, but Fulham have managed to lure him to England.
Grimmer's current deal expires at the end of the season and Fulham was ready to sign the midfielder on a pre-contract.
But Grimmer is thought to have travelled to London today to finalise his move to Craven Cottage during the January transfer window rather than wait until he sees out his contract in the summer.
The fee is likely to be in the region of £250,000 plus add-ons.
STV's Chris Harvey has Tweeted: "Jack Grimmer signs for #Fulham FC, leaves #Aberdeen after coming through the ranks at Pittodrie."
And Grimmer cryptically Tweeted today: "Thanks for all the good messages! Here we go :)"
Grimmer became the youngest player in Aberdeen's history when he made his debut against Rangers in April 2010 aged 16 and two months.