The Senegalese striker has scored three goals in all competitions so far this season in Spain, including two goals and two assists in La Liga.
The 21-year-old - who was also reportedly a target for Southampton - can play as a striker or winger on either side.
He has been capped once by Senegal, making his debut after coming on as a substitute in his nation's 2-0 defeat by the Netherlands at the World Cup in Qatar.
The forward has been out of action for two weeks due to a muscle injury but is close to returning to full fitness.
Jackson will become Gary O'Neil's second signing of the January transfer window following Dango Ouattara's £20m move from French side Lorient.
Bournemouth currently sit in 18th in the Premier League table, two points above the bottom.