The Championship side have been heavily linked with the Wales international, who has netted 21 goals in 93 appearances for the Tykes over the past two seasons.
The Lions have now revealed that they have completed a deal for Bradshaw, who will initially join on loan before costing a club-record fee in January.
Manager Neil Harris told the club's official website: "He's a player who I think can excel at our football club, and he certainly has something different to our front line with how he plays.
"He makes excellent use of the ball, he's a link player, but certainly a poacher in the penalty area as well. I'm very much looking forward to welcoming him to compete with our other attacking options."
Bradshaw could feature when Millwall travel to Rotherham United this weekend.