The Grizzlies announced the moves Monday.
Jackson was the fourth overall draft pick in 2018, and he averaged 13.8 points and 4.7 rebounds in 58 games to earn NBA All-Rookie First Team honors. At 19, Jackson was the second-youngest player in the NBA last season. He turned 20 last month.
The Grizzlies acquired Allen in a trade with Utah that sent Mike Conley to the Jazz this summer. The 6-foot-3 Allen was the 21st overall pick in 2018 after playing four years at Duke. He averaged 5.6 point in 38 games for Utah and had a career-high 40 points against the Clippers on April 2, the second-highest scoring game by a rookie last season.