Shamrock Rovers Having come through the academy at
Shamrock Rovers, Bazunu made his debut for Rovers' first team on 9 June 2018 in a 5–0 win at home to
Bray Wanderers, at the age of 16. Manager
Stephen Bradley stated that despite not wanting to 'throw [young goalkeepers] in that early', Bazunu was 'ready' for his first-team debut. Bazunu also made two appearances in the
2018–19 UEFA Europa League.
Manchester City On 6 September 2018, it was announced that Bazunu had joined
Premier League club
Manchester City for an undisclosed fee reported to be £420,000, though a joining date had not been set, with Bazunu wishing to complete his education in the Irish education system. The following month, it was announced he has signed a pre-contract agreement with Manchester City, with it initially agreed he would join the club in summer 2019, before it was moved forward to February 2019. He made his debut for Manchester City's
under-18 side on 9 February 2019 against
Stoke City U18.
Loans to Rochdale and Portsmouth He signed a contract extension with Manchester City until 2024 and joined
League One side
Rochdale on a season-long loan later that month. He started and kept a clean sheet on his debut for Rochdale on 5 September 2020 in a 1–0
EFL Cup victory away to
Huddersfield Town, and made his first league appearance the following week in a 3–1 defeat at
Swindon Town. On 1 July 2021, he moved on loan to
Portsmouth for the 2021–22 season. He was named as the club's Players' Player and Player of the Season.
Southampton On 17 June 2022, Bazunu joined
Southampton on a five-year contract for a reported fee of £12 million. On 6 August 2022, he made his first
Premier League appearance for Southampton in a 4–1 defeat to
Tottenham Hotspur. Bazunu got his first clean sheet in the Premier League and for Southampton on 19 October 2022 in a 0–1 victory against
Bournemouth. Ahead of a match against
Preston North End on 16 April 2024, he suffered an achilles injury during the warm-up. Manager
Russell Martin subsequently confirmed that Bazunu would be sidelined for nine to ten months. On 10 January 2025, he played for the under-21 side in a 5–0 victory over
Fulham in Premier League 2.
Standard Liège (loan) On 3 February 2025, he joined
Belgian Pro League club
Standard Liège on loan for the remainder of the 2024–25 season. On 24 April 2025, it was announced that Bazunu's loan was cut short due to a knee injury and he would be returning to Southampton.
Stoke City (loan) On 13 January 2026, Bazunu joined
Stoke City on loan for the remainder of the
2025–26 season. ==International career==