Honda team boss Nick Fry considers the Briton as one of Formula One's best drivers.
Three times world champion Jackie Stewart said the Briton can go on to become a multiple race winner and world champion.
BAR said they took the decision to take up the option on Button's contract on July 20 and informed the driver's management team.
Williams said the 24-year-old had signed for the team for the 2005 season and beyond.
The 28-year-old believes BAR, and not Williams, offer him the best chance of becoming Britain's next F1 champion.
Although Ralf Schumacher, last year's pole-setter in a Williams, qualified second fastest the German lost 10 places on the grid after having an engine replaced on Thursday.
Ferrari will present their new Formula One car online on January 30, a day after McLaren take the wraps off their Honda-powered challenger.
The Ferrari driver believes Sunday's Spanish Grand Prix will not be the same without the excluded BAR team and Jenson Button.
Kimi Raikkonen's engine caught fire on Friday, forcing him to replace the unit and start Sunday's race 10 places down from wherever he qualifies on Saturday.
The Italian grabbed the third pole position of his career, knocking Honda's Jenson Button off the top spot as the seconds ticked away at the end of the session.
BAR, runners-up to Ferrari in the championship last season, have yet to score a point in two races.
The Honda-powered team reached an agreement with Bridgestone to end their partnership.
Jenson Button's BAR Formula One team has been put up for sale -- but only to allow a new joint venture between engine partners Honda and majority owners British American Tobacco to buy it back.
BAR have called the 24-year-old to a face-to-face meeting with the team by Monday evening or he will face a legal fight over his move.
Ferrari's world champion Michael Schumacher, winner at Silverstone last year, qualified a disappointing ninth.