*Club domestic league appearances and goals, correct as of 16:16, 13 April 2024 (UTC) ‡ National team caps and goals, correct as of 06:20, 27 March 2024 (UTC)
Having come through the academy at Shamrock Rovers,[7] 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.[2][8] Manager Stephen Bradley stated that despite not wanting to 'throw [young goalkeepers] in that early', Bazunu was 'ready' for his first-team debut.[8] He made three further appearances for Shamrock Rovers in the 2018 League of Ireland Premier Division.[2] These included a penalty save at Cork City which meant he kept four clean sheets out of four in the league.[9] Bazunu also made two appearances in the 2018–19 UEFA Europa League.[10]
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.[11][12] The following month, it was announced he has signed a pre-contract agreement with Manchester City,[13] with it initially agreed he would join the club in summer 2019, before it was moved forward to February 2019.[14] He made his debut for Manchester City's under-18 side on 9 February 2019 against Stoke City U18.[14]
He signed a contract extension with Manchester City until 2024 and joined League One side Rochdale on a season-long loan later that month.[16][17] 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,[18] and made his first league appearance the following week in a 3–1 defeat at Swindon Town.[19]
On 1 July 2021, he moved on loan to Portsmouth for the 2021–22 season.[20][21] He was named as the club's Players' Player and Player of the Season.[22]
Southampton
On 17 June 2022, Bazunu joined Southampton on a five-year contract[23] for a reported fee of £12 million.[24] On 6 August 2022, he made his first Premier League appearance for Southampton in a 4–1 defeat to Tottenham Hotspur.[25] Bazunu got his first clean sheet in the Premier League and for Southampton on 19 October 2022 in a 0–1 victory against Bournemouth.[26]
Ahead of a match against Preston North End on 16 April 2024, he suffered an achilles injury during the warm-up and was expected to miss the remainder of the 2023–24 season.[27] Manager Russell Martin later confirmed Bazunu would be sidelined for nine to ten months.[28][29]