His first professional club was Serbian Mladi Radnik Pozarevac. In 2011 he moved to Polish Champion, one of the most successful Polish team of PlusLiga, PGE Skra Bełchatów. With PGE Skra Belchatów he won silver medal of Polish Championship in 2011/2012. He has silver medal of Club World Championships from 2012. He won with the club from Bełchatów the Polish Cup in 2012. In 2012 PGE Skra Belchatów, with Konstantin, gained the silver medal of CEV Champions League after the match against Zenit Kazan on Final Four in Łódź, Poland. The match ended controversially, because the judge didn't see an error by a Russian player and ended the match despite the fact that the audience and the players saw the error on screen. He was a player of Sir Safety Perugia in season 2013/2014. On May 4, 2014, he won silver medal of Italian Championship 2013/2014. On May 8, 2014 was announced officially that Čupković moved to the Polish club Transfer Bydgoszcz. In 2015 went to Tours VB. On May 25, 2016, his transfer to Olympiacos was announced officially.