The south coast club, who face high-flying Liverpool on Saturday, said scans on Wednesday revealed the 21-year-old had ruptured the anterior cruciate ligament in his right knee.
"We are all devastated for Lewis," Bournemouth manager Eddie Howe told the club website on Thursday.
"It is a huge blow to a very talented player who has been a very important part of our side over the last 18 months.
"But Lewis is a strong character who only has to look around our dressing room for inspiration of how an injury like this is merely a setback and to know that he can come back better than ever."
Cook, an England under-21 captain who joined Bournemouth from Leeds United in July 2016, has made 15 appearances for the club this season.
Bournemouth are seventh in the league, behind Everton and ahead of Manchester United on goal difference.
(Reporting by Alan Baldwin, editing by Keith Weir)
