A product of FC Barcelona's La Masia, he played for their reserves before transferring to Red Star Belgrade in January 2016, winning the Serbian SuperLiga in his first season.

[2] Babunski was promoted to the reserves in June 2013,[3] and made his professional debut on 24 August, coming on as a second-half substitute in a 2–1 home win against CD Lugo in the Segunda División championship.

On 28 February 2015, as a half-time substitute for Juan Cámara, he received a straight red card in a 2–4 loss to RCD Mallorca at the Mini Estadi.

[8] He made his Serbian SuperLiga debut on 20 February, replacing Aleksandar Katai for the final 11 minutes of a 2–1 home win over FK Mladost Lučani,[9] and played five more matches as the team won their 27th title.

[10] On 30 January 2017, Babunski signed for Japanese club Yokohama F. Marinos,[11] as their second purchase from Red Star that month after that of Hugo Vieira.

[28] He, his brother, and four friends spent Christmas 2015 at a refugee camp in Gevgelija, a Macedonian town on the Greek border and part of the Balkan route for displaced people seeking accommodation in the West.