Madsen discovered football on a vacation in Sweden at age 4, and shortly afterwards began playing for Gl.
[13] His debut in the Austrian Bundesliga came less than a month later, on 19 September in a 1–1 away draw against WSG Swarovski Tirol where he came on as a second-half substitute for James Holland.
[15] On 26 November, Madsen made his first European appearance in the group stage of the 2020–21 UEFA Europa League, playing as a starter in the 0–2 home loss to Royal Antwerp.
[2] On 5 June 2021, Madsen signed a four-and-a-half-year deal with Czech First League club Slavia Prague.
[20] On 1 July 2022, Madsen returned to his homeland, as he signed a deal until the end of 2026 with Danish Superliga club AGF.
[21] After a great start to the 2024-25 season with goals and assists, Madsen was named player of the month in the Danish Superliga in August.