Jackson played a vital role for Villarreal last season, scoring nine goals in the final eight matches and winning the LaLiga Player of the Month award for May.
Jackson scored 12 goals in 26 league matches last term. He also made his debut for Senegal at the World Cup in Qatar, playing for 16 minutes in the 2-0 defeat by the Netherlands.
Chelsea did not give any financial details but British media said they will pay 37 million euros for Jackson.
Following Christopher Nkunku's (25) arrival from RB Leipzig, he is the second forward to join Chelsea this month.
The Stamford Bridge side will seek to revive their fortunes under former Tottenham Hotspur manager Mauricio Pochettino after finishing 12th - their worst finish to a season since 1994.