The 21-year-old has made an initial loan move until the end of the season, with a view to a permanent transfer.
Davis arrives following the departure of Diego Rico to Real Sociedad.
After signing for the Cherries, Davis spoke to the club's website, saying: "I'm absolutely delighted, when I got told the club was interested I spoke to my agent and said I'd be more than happy to come down.
"Now I'm down here I'm delighted to be here and I want to try and help the club get back to the Premier League because that's where they belong.
"The stadium's beautiful, I can't wait to see all the fans in."