Born in England to a British mother and a Montenegrin father, Sarkic represented Montenegro at youth and senior international level.

In April 2017, he was named on the bench for a league match against Reading as cover for Sam Johnstone following an injury to Mark Bunn.

[8] Šarkić joined Wigan Athletic on 31 August 2017 on a season-long loan,[9] going on to make 3 appearances, all in cup competitions, and saving a penalty on his debut against Middlesbrough U21 in the EFL Trophy.

[11] On 26 June 2019, Sarkic signed for Scottish Premiership club Livingston on what was intended to be a season-long loan,[12] but he was recalled by Aston Villa on 3 January 2020, as their first-choice goalkeeper Tom Heaton had suffered a knee injury.

[18] He kept 11 clean sheets in 29 games across all competitions for the Shrews before returning to Wolves on 6 May 2021; he had competed for a starting position against Harry Burgoyne during his loan.

[20] With previous first-choice goalkeeper Neil Etheridge still recovering from COVID-19, Sarkic started the opening-day fixture away to Sheffield United and kept a clean sheet as Birmingham won 1–0.

[21] He retained his place in the starting eleven even after Etheridge's return to fitness, kept ten clean sheets, and missed only one of Birmingham's 24 league matches.

[24] Although Wolverhampton Wanderers' goalkeeping coach, Tony Roberts, thought Sarkic's progress would be better served by playing regularly on another loan,[25] he signed a new three-year contract, and technical director Scott Sellars said that he would remain at the club as backup and competition for José Sá.

[4][42] On 5 June 2024, Sarkic played his final game, starting in a 2–0 friendly defeat to Belgium; on the occasion, he was named man of the match for his performance,[37] which included nine saves.

[43] In the early morning of 15 June 2024, Sarkic collapsed at an apartment in Budva, Montenegro; his long-term partner Phoebe, former Aston Villa teammate Oscar Borg and his girlfriend called an ambulance.