Born in Bradford, West Yorkshire, Tavernier joined the Leeds United youth system at the age of nine after being picked up from local club Farsley Celtic.

[20] Tavernier made his European debut on 23 August 2012, playing the entirety of Newcastle's 1–1 draw at Atromitos in UEFA Europa League qualification.

He totalled seven appearances for the team across four competitions that season, making his Premier League debut on 29 September by coming on for Steven Taylor in the 56th minute of a 2–2 draw away to Reading.

[21] Tavernier joined Shrewsbury Town on 26 July 2013 on an initial month-long loan,[22] but returned to his parent club early having aggravated a pre-existing metatarsal injury.

[28] He made his Championship debut on 9 August, replacing Don Cowie for the final 18 minutes of a 2–2 draw with Reading at the DW Stadium in the first game of the season.

[32] On 20 July 2015, Tavernier and Wigan teammate Martyn Waghorn both signed three-year contracts with the then Scottish Championship club Rangers for a transfer fee of £200,000.

[37] In October, having scored nine goals by that stage in the season, Tavernier admitted that he and Waghorn had been short of confidence before moving to Rangers, due to the inconsistency and squad rotation at Wigan.

[5] On 5 April 2016, he scored the only goal of the home win over Dumbarton that gave the team the title and promotion back to the top flight after a four-year absence.

[38] Five days later in the Challenge Cup Final at Hampden Park, he netted a long-range volley in the first half of a 4–0 victory against Peterhead, and was named man of the match.

[39] On 17 April, in a Scottish Cup semi-final against Old Firm rivals Celtic, he missed in the penalty shootout after a 2–2 draw, but Rangers nonetheless won 5–4.

[40] After a strong first season, Rangers offered Tavernier a new contract which he rejected, with media reports citing the financial rewards were not good enough.

[48] Cameron Mack, a 21-year-old man from Port Seton in East Lothian, jumped over the advertising boards and approached Tavernier before being led away by police and stewards and arrested.

[59] He netted another brace on 14 April, as the team won 3–1 at home to Braga after extra time to make the semi-finals,[60] where he scored against RB Leipzig.

He was the top scorer with seven goals as the team finished runners-up to Eintracht Frankfurt and netted Rangers' first attempt in the penalty shootout loss in the final.