[citation needed] The Baltimore Orioles selected Bundy in the first round, with the fourth overall pick, in the 2011 Major League Baseball draft.

[9] Bundy made his professional debut on April 6, 2012, with the Single-A Delmarva Shorebirds pitching against the Asheville Tourists.

[16] On September 23, Bundy made his MLB debut against the Boston Red Sox in a relief appearance.

[17] In January 2014, Bundy had circled June 28, 2014—one year and one day after his Tommy John surgery—as the target date for his return.

On August 26, it was announced that he would pitch in an instructional league, along with fellow Orioles pitcher Hunter Harvey, in September 2015.

[20] Out of minor league options in 2016, Bundy was impressive in spring training and made the Orioles Opening Day roster.

He made his season debut on April 7, 2016; this appearance came 1,290 days after the last time Bundy had played in a Major League game.

On July 27, against the Colorado Rockies, Bundy took a perfect game into the sixth inning with one out before allowing a walk and then a home run to former Oriole Nick Hundley.

In his very next start against the Texas Rangers, Bundy turned in his best performance, throwing 7 innings of one-hit baseball.

[21] In 2017, Bundy was slotted as the Orioles number two starter after injury to staff ace Chris Tillman.

Bundy struggled in the month of June and July, and finished the first half with an ERA of 4.33 in 108 innings of work.

On August 1, he had the longest outing of his career, going eight innings against the Kansas City Royals, allowing one unearned run in the Orioles 7–2 victory.

Bundy gave up the home runs to Jorge Soler, Mike Moustakas, Salvador Perez and Alex Gordon.

In the 4-3 no-decision win against the Chicago White Sox, Bundy pitched six innings, striking out six batters and allowing three runs.

[28] On June 28, 2021, in a game against the New York Yankees, Bundy suffered a bout of heat exhaustion and vomited on the field.

[29][30] After going 1–7 with a 6.78 ERA in his first 14 starts, Bundy was replaced in the rotation by José Suárez and was moved to the bullpen on June 30.

[31] On July 1, he became the second established member of the Angels starting rotation to be moved into the bullpen during the season, joining José Quintana.

[32] On December 1, 2021, the Minnesota Twins signed Bundy to a one-year contract with a club option for the 2023 season.