Born in Freiburg im Breisgau, Germany, to a German father and Tunisian mother, Dräger played his youth football with local Freiburg Police Sports Club and in 2009 he moved to the youth department of SC Freiburg. With the club's A youth team, he won the DFB-Pokal youth Cup in 2014. They won the final with an 8–7 win on penalties against FC Schalke 04, with Dräger being one of the successful penalty takers. After being the most dangerous player in the Freiburg youth team in the 2014/15 A-Junior Bundesliga with 15 goals during that season, he advanced to their second team in 2015. In the 2016/17 season, they managed direct promotion back to the Regionalliga Südwest as the Baden-Württemberg league champion. Dräger played mainly for the second team, but he made his professional debut for SC Freiburg on 27 July 2017, in a UEFA Europa League qualifying match against Slovenian club Domžale, coming on as a substitute in the 87th minute for Mike Frantz.[4]
In 2018, Dräger joined SC Paderborn on a two-year loan.[5] In September 2020, he joined Greek club Olympiacos on a transfer fee in the range of €1 million.[6]
On 31 August 2021, Dräger joined EFL Championship side Nottingham Forest for an undisclosed fee.[7] On 2 February 2022, Dräger moved on loan to Swiss Super League club Luzern, with an option to buy. After the loan period, Luzern did not pull the option.[8]
On 25 August 2023, FC Basel announced they had signed Dräger on a three-year contract.[9] He joined Basel's first team for their 2023–24 season under head coach Timo Schultz. Dräger played his domestic league debut for the club in the home game in the St. Jakob-Park on 3 September as Basel played a 2–2 draw with Zürich.[10] He scored his first goal for his new team in the home game on 5 November 2023 as Basel won 2–1 against Yverdon.[11]
^FC Basel 1893 (25 August 2023). "Mohamed Dräger zum FCB" [Mohamed Dräger to FCB] (in German). FC Basel 1893 AG. Retrieved 25 August 2023.{{cite web}}: CS1 maint: numeric names: authors list (link)
^Verein "Basler Fussballarchiv” (3 September 2023). "FC Basel - FC Zürich 2:2 (0:1)". Verein "Basler Fussballarchiv”. Retrieved 16 November 2023.