Morris was born in Grand Rapids, Michigan and his mother is Lationa Morris. His mother nicknamed him "Man-Man" when he was born. He grew up in Flint, Michigan attending Flint Beecher High School. He was a four-year starter for the Beecher Buccaneers. Morris was a three-time winner of Michigan's Associated Press Class C Player of the Year Award and a three time all-state selection. He led the Buccaneers in scoring, assists, and steals in all four seasons. He led Flint Beecher to back-to-back Class C Michigan state titles in 2012 and 2013. One of the nation's best point guards, Morris won Michigan's Mr. Basketball award in 2013.
He was ranked No. 96 in the final Rivals.com national rankings and No. 89 in the Scout.com ratings as well as being a Parade All-American. Monte was recruited to Butler, Illinois, Indiana, Arizona State, Cincinnati, Georgia Tech, USC, and Iowa State which he ultimately committed to.
Morris started the season on the bench but worked his way into the starting line-up against the Oklahoma Sooners February 1, 2014 staying there ever since. Morris averaged 6.8 points, 3.7 assists and 2.6 rebounds and dished out 134 assists, the third most by any freshman in school history. He shot 84.7 percent (61-72) at the free-throw line, the second-best mark by an ISU freshman and was the only Big 12 freshman to make at least 25 threes (28-69) and shoot over 40 percent. He scored in double figures in 13 games, including all three games in the NCAA Tournament. Morris tallied a season-high 15 points in the Cyclones' NCAA Tournament win against NCCU and followed that with 13 against North Carolina Tar Heels. He dished out five or more assists in 12 games and led all Big 12 freshmen with 46 steals, the fifth most by an ISU rookie. His 1.3 steals per game ranked seventh in the Big 12 and he had an ISU freshman record 12 assists with no turnovers against West Virginia. He played 52 minutes of turnover free basketball in ISU's triple-OT win at Oklahoma State and had 10 points and five assists, including the game-winning 3-pointer with 43 seconds left in the third overtime against the Cowboys. He broke the NCAA record and led the nation with a 4.79 assist-to-turnover ratio, including a 6.9 assist-to-turnover ratio in Big 12 play.
Monte started all 34 games and averaged 11.9 points, second on the team. He also posted a Big 12-best 5.2 assists and 3.4 rebounds on his way to earning All-Big 12 Second-Team honors. He had second-most assists of any sophomore and 11th-most by any player in school history. Monte was second in the Big 12 in field goal percentage shooting 50.7 percent from the field, had 64 steals, the second-most by a sophomore and tied for the ninth-most by any player in a single season in school history. His 110 steals in the last two seasons were the most by any Big 12 player. His first-career 20-point game came against Texas Tech, he recorded his first double-double and just missed a triple-double with 11 points, 10 assists and seven rebounds against Kansas. He tallied a career-high 24 points and hit buzzer-beating 17-foot fadeaway to beat Texas in Big 12 quarterfinals. He averaged 15.3 points to earn Big 12 All-Tournament team honors as the Cyclones won the Big 12 Tournament for the second consecutive season and he did not commit a turnover in three games at the Big 12 Championship. He led the nation for the second consecutive season with a 4.63 assist-to-turnover ratio and finished season with 176 assists and just 38 turnovers.
On February 1, 2016, he was named one of 10 finalists for the Bob Cousy Point Guard of the Year Award. He was named to the 35-man midseason watchlist for the Naismith Trophy on February 11.
Morris received preseason recognition on November 2, 2016, as the Associated Press named him to the preseason All-American team. At the end of his senior season, Morris broke the NCAA record assist-to-turnover ratio from 4.79 he set his freshman year to the new record of 5.21. In 2017 Morris lead Iowa State to a 5th seed and got to the second round before losing to Purdue
Morris was selected in the second round, 51st pick overall by the Denver Nuggets in the 2017 NBA draft. Following the Summer League he signed a two-way contract with the Nuggets. Under the terms of the deal he will split time between the Nuggets and a G League that would be best designated for him.