[3] He began attracting the interest of several clubs, including Red Star Belgrade, Anderlecht, Udinese, and Tottenham Hotspur.

[15] After the relegation of VfB Stuttgart at the end of the season 2015–16, Kostić was transferred to Hamburger SV for a fee of €14 million, making him the most expensive player in the history of the club.

On 14 April 2022, Kostić scored two goals as Frankfurt defeated Barcelona 3–2 at the Camp Nou and 4–3 on aggregate in the UEFA Europa League to qualify for the semi-finals.

[18] Eintracht later won the Europa League, and Kostic was pronounced player of the season[19] and was also top assists provider.

[24] On 7 June 2015, Kostić debuted for the Serbian senior squad against Azerbaijan in a 4–1 friendly victory in Sankt Pölten, Austria, making way for Lazar Marković after 56 minutes.

[27] In June 2018, he was selected in Serbia's squad for the 2018 FIFA World Cup,[28] playing all three group stage matches.

[35] Kostić was part of the Serbia squad for UEFA Euro 2024 and started the team's opening match of the tournament against England.