On his debut for Leipzig, a home Bundesliga game against Borussia Dortmund on 10 September, Burke came on after 69 minutes in place of Yussuf Poulsen and provided an assist for Naby Keïta to score the only goal. Fifteen days later he opened the scoring in a 1–1 draw at 1. FC Köln, becoming the first Scot to score in Germany’s top flight since Brian O’Neil for VfL Wolfsburg in November 1999.

Burke was born in Kirkcaldy, Scotland, but grew up in Melton Mowbray, Leicestershire, where he joined local club Mowbray Rangers aged eight. After one season at the club, he was scouted to join Nottingham Forest’s Academy where he progressed for the next nine years. Burke signed his first professional contract with Nottingham Forest on 19 September 2014, before making his professional debut as an 87th-minute substitute during a 3–1 defeat against Tottenham Hotspur on 24 September. He made his league debut on 14 February 2015 as a 64th-minute substitute in a 4–4 draw away to Blackpool.

Burke was eligible to play for England in addition to Scotland, having been brought up south of the border from a young age, but he opted to play for his country of birth. He was first called up by Scotland U19 and scored on his debut for the team, a 3–1 win against Mexico U18, on 6 October 2015. He went on to make six appearance for the side.

Ahead of the 2015–16 season, Burke signed another three-year contract with the club on 29 July. Burke played for the club’s reserves until December when he was called up to the first team by manager Dougie Freedman, who Burke considered his mentor. On 29 December, Burke scored his first goal for Forest from a curling shot hit with the outside of his right foot against Cardiff City. His goal opened the scoring at the Cardiff City Stadium nine minutes into the game, which ended 1–1. Burke scored his second goal for the club on 16 January 2016 in a 3–0 win over Bolton Wanderers. His performance once again earned a contract extension on a four-year deal on 29 February. Burke ended the 2015–16 season having made twenty-one appearances in all competitions and scoring twice.

On 28 August 2016, in a move that caused outrage among Forest supporters, who criticised the club’s owner Fawaz Al-Hasawi, Burke joined newly promoted Bundesliga club RB Leipzig on a five-year contract for an undisclosed fee. The deal was reported by BBC Sport to be worth around £13 million, which set a new record for a transfer involving a Scottish player.

Burke received his first call-up to the senior Scotland squad on 10 March 2016 for their friendly match against Denmark. He made his debut as an 82nd-minute substitute for goalscorer Matt Ritchie in the 1–0 win at Hampden Park 19 days later. He was left out of the full squad in June 2017 and was instead selected for an under-20 squad in the 2017 Toulon Tournament. In the first match there, Burke scored both Scotland goals in a 3–2 defeat against the Czech Republic.

Burke returned to English football on 25 August 2017, moving to Premier League side West Bromwich Albion for a reported fee of £15 million, breaking his own record as the most expensive Scottish footballer. He signed a five-year contract at the West Midlands club. Burke said that he chose West Bromwich Albion as a way to have more game time, even though with Leipzig he could play in the Champions League.

On 14 August 2018, Burke scored his first goal for the Baggies, the game’s only in a home win over Luton Town in the first round of the EFL Cup. He made only six total appearances in the first half of the season, of which three were in the Championship, all from the bench.

During the 2017–18 season, Burke was selected in the under-21 squad. He made his debut at that level on 5 September, and he scored the first goal in a 2–0 win against the Netherlands. In the 2018 Toulon Tournament, Burke scored goals against France and South Korea as Scotland finished in fourth place. In the third-place play-off against Turkey, he was the only player to miss in the penalty shootout defeat, with Altay Bayındır saving his attempt.

Burke moved on loan to Celtic on 5 January 2019 until the end of the season. He made his debut two weeks later, playing the full 90 minutes of a 3–0 home win over Airdrieonians in the Scottish Cup. On 23 January, his Scottish Premiership debut, he played as a striker in the absence of Leigh Griffiths and Odsonne Édouard and netted twice in a 3–0 win at Celtic Park, his first league goals since September 2016.

On 30 August 2019, Burke was loaned to La Liga club Alavés for the season. Burke made his debut for his new club in their 1–0 defeat to Sevilla on 15 September. He scored his first goal for Alavés in a 2–1 La Liga win over Eibar on 7 February 2020.

Burke was recalled to the full international squad in March 2019 and scored his first senior goal in Scotland’s 2–1 victory over Cyprus on 8 June the same year.

Sheffield United signed Burke from West Brom in September 2020, with Callum Robinson moving in the opposite direction. Burke signed a three-year contract with the Blades.