Philadelphia 76ers (2014–2016)
On June 26, 2014, Grant was selected with the 39th overall pick in the 2014 NBA draft by the Philadelphia 76ers, and joined the team for the 2014 NBA Summer League. On September 29, 2014, he signed a four-year deal with the 76ers, with two years guaranteed. On January 21, 2015, he recorded four points and eight blocks in a loss to the New York Knicks. The eight blocks were the most in a game for a 76er since Samuel Dalembert had nine on December 12, 2007, and the most by a 76ers' rookie since Shawn Bradley had nine on January 17, 1994. On February 2, he had a season-best game with 18 points and 7 rebounds in a loss to the Cleveland Cavaliers.
In July 2015, Grant re-joined the 76ers for the 2015 NBA Summer League. On November 11, 2015, he recorded his first career double-double with 12 points and 10 rebounds in a loss to the Toronto Raptors. On December 30, he recorded 16 points, a career-high 11 rebounds and five blocked shots in a 110–105 win over the Sacramento Kings.
Oklahoma City Thunder (2016–2019)
On November 1, 2016, Grant was traded to the Oklahoma City Thunder in exchange for Ersan İlyasova and a protected draft pick. He made his debut for the Thunder the following day in an 85–83 win over the Los Angeles Clippers, recording six points, two rebounds and two blocks in 18 minutes off the bench. On December 19, 2016, he scored a season-high 15 points in a 110–108 loss to the Atlanta Hawks. On February 1, 2017, he tied his season high mark with 15 points in a 128–100 loss to the Chicago Bulls.
On October 31, 2017, Grant scored 17 points off the bench in a 110–91 win over the Milwaukee Bucks. He finished 5 of 9 from the floor to record 17 points for the first time since April 1, 2016.
On July 7, 2018, Grant re-signed with the Thunder. On January 10, 2019, he scored a then career-high 25 points to go with 12 rebounds in a 154–147 double-overtime loss to the San Antonio Spurs. On March 18, he scored 27 points in a 116–107 loss to the Miami Heat. On April 10, he set a new career high with 28 points in a 127–116 win over the Bucks.
Denver Nuggets (2019–present)
On July 8, 2019, Grant was traded to the Denver Nuggets for a 2020 first-round pick.