He spent time on loan at Swindon Town, where he made his senior debut, Burton Albion and Sparta Prague.
FC Slovácko in his native Czech Republic, but moved to England to join Manchester United in January 2018 at the age of 17.
[5] With the first team's qualification for the knockout phase of the 2019–20 UEFA Europa League secure, manager Ole Gunnar Solskjær selected a young team for their matchday 5 game against Astana in Kazakhstan on 28 November 2019; Kovář was named on the bench as back-up for Lee Grant, but did not play in the match.
[9] Kovář was named in United's final Premier League squad submitted after the end of the transfer deadline for the 2021–22 season.
[10] On 31 January 2022, Kovář signed a new contract with United until June 2023 and moved on loan to Burton Albion until the end of the season.
[22] Kovář was first called up to the Czech Republic senior team on 8 September 2021 for a friendly match against Ukraine after Tomáš Vaclík suffered an injury.