Madsen started playing football at the age of three in local club Marienlyst and later joined Næsby. In November 2014 it was confirmed that Madsen, who already earlier had been on trials at Midtjylland and also Brøndby, would join Midtjylland from the summer 2015.[1] On his 15th birthday, he signed a 3-year youth contract with the club.[2][3] In 2017, he extended his contract until the summer 2019.
After returning to Midtjylland from his loan spell, Madsen was sold to Belgian football; K.V.C. Westerlo confirmed on 20 July 2022, that Madsen had signed a deal until June 2026.[7]