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Doak was left out of Liverpool's 4-2 win over Bournemouth at Anfield on Friday night, amid ongoing talks between the two Premier League clubs.
But Bournemouth have since secured a deal for the forward, who has six caps for Scotland, and spent last season on loan at Middlesbrough, scoring three times and recording seven assists in 24 matches.
They are understood to have beaten interest from other Premier League clubs, including Leeds, as well as Portuguese club Porto.
Doak's move will give him a better chance of playing first team football having played just 10 times for Liverpool since joining from Celtic in 2022.
Bournemouth have had a busy summer having sold Dango Ouattara to Brentford for £42.5m last week.
They already lost defenders Illia Zabarnyi, Dean Huijsen and Milos Kerkez for a combined total of about £150m to Paris St-Germain, Real Madrid and Liverpool respectively.
Bournemouth manager Andoni Iraola said his side are "not where they want to be" before defeat at Anfield and added "important movements" will be made before the transfer window closes on 1 September.
Goalkeeper Djordje Petrovic, left-back Adrien Truffert and centre-back Bafode Diakite have joined the club this summer, with January signing Eli Junior Kroupi also coming into the squad after finishing last season on loan at Lorient.
Sources say the Cherries are in the market for a further central defender and a striker to compete with Evanilson, with alternative option Enes Unal expected to be out with a major knee injury until November or December.
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