The Scouse stopper has been at the club since he was nine years old. He made his u-18s debut as an u-16 player in the 2019/20 season in the FA Youth Cup against Tottenham. Despite having not even played at Under 23 level, Davies was on the first-team bench when Liverpool played Red Bull Leipzig in the last 16 of the Champions League on 10 March 2021. He was also included in the squad for both legs of the quarter final against Real Madrid. 

In 2021/22 Davies featured primarily for the under 18's but was on the bench for a Champions League group game away to AC Milan. He was on the bench for the Community shield victory over Manchester City in August 2022 as well as three Premier League games that month. He played twice in the EFL Trophy and was the most used keeper for the Under 21's in Premier League 2 during 2022/23. A season long loan deal was agreed for Davies to spend 2023/24 on loan at League Two side Crewe Alexandra, where he made 32 appearances in all competitions. He remained at Liverpool for 2024/25, making 10 Premier League 2 appearances and being on the bench four times in the Champions League. Davies signed a new contract in June 2025 and a loan move was arranged to spend 2025/26 at Crawley Town in League Two.