Bajrami was born in Birmingham in 1999, just two weeks after his Albanian parents left Kosovo because of the war. He was raised in the Quinton district and attended Priory School in Edgbaston. He had two unsuccessful trials with Birmingham City's Academy before he was accepted at under-14 level, and took up a two-year scholarship in July 2016. Academy coach Steve Spooner described him as "quite an unassuming young man but is very much a leader and is very popular with the rest of the players."