[1][2] Glumac started his career playing at youth level for his hometown club GOŠK Dubrovnik.
[3][4] In the 2011–12 season, Glumac returned to Hajduk and made his debut for the first team on 21 October 2011 as a late substitute in the 3–0 win against Lučko.
[6] On 9 January 2012, it was announced that Glumac was transferred to RNK Split in exchange for Goran Milović, signing a two-and-a-half-year contract.
[7] Glumac was a part of the Croatian national under-19 team in the 2010 UEFA European Under-19 Football Championship held in France.
Glumac was selected in the 21-man squad by manager Ivica Grnja despite initial objection of Hajduk as they wanted to keep the defender in case of first-team injuries.