Davies started his career with the Everton youth system at the age of 14. He started his scholarship with Everton in July 2008 and broke into the under-18 team during the 2008–09 season. He was named as the AcademyPlayer of the Season for 2009–10. Davies joined the first team on a pre-season tour of the United States in July 2011 before playing regularly for the reserve team in 2011–12. He was released by Everton at the end of the season. Davies signed for Championship club Sheffield Wednesday on 23 October 2012 on a month-to-month contract as backup to first-choice goalkeeper Chris Kirkland. He was an unused substitute for the first team on multiple occasions throughout the 2012–13 and 2013–14 seasons before being released by the club.
Barnsley
Davies signed for League One club Barnsley on 13 June 2014 on a two-year contract. He made his first-team debut on 9 August 2014 in the opening match of 2014–15, a 1–0 home defeat to Crawley Town. Davies played 25 times in 2014–15 as the Tykes were unable to mount a promotion challenge finishing in 11th. He signed a new contract with Barnsley in May 2015. Barnsley had a successful 2015–16 season as they recovered from a slow start to gain promotion via the play-offs, beating Millwall 3–1 in the 2016 Football League One play-off Final. They also beat Oxford United 3–2 in the 2016 Football League Trophy Final. On 26 January 2017, Davies signed a new contract with Barnsley until June 2019. Davies was a ever-present in 2016–17 helping the side finish in a safe mid-table position of 14th. He was appointed as the team's vice-captain on 3 August 2017. He was later named first team captain. Barnsley were relegated from the Championship in 2017–18. Davies was one of the key players in Barnsley's 2018–19 season, during which the Tykes won promotion back to the Championship, starting 42 of their 46 games. He made vital contributions in crucial fixtures, most notably the away fixtures against Doncaster Rovers and Portsmouth, saving a second-half penalty in the latter. On 26 March 2019, Davies was one of four Barnsley players named in the EFL Team of the Season for League One. Davies was also named in the overall EFL Team of the Season on 8 April 2019. Davies' season was again recognised on 24 April 2019 when he was named in the PFA Team of the Year for League One along with four teammates. He was offered a new contract by Barnsley at the end of the 2018–19 season.
Stoke City
Davies signed for Championship club Stoke City on 25 June 2019 along with Barnsley teammate Liam Lindsay. He made his debut for Stoke in an FA Cup tie against Brentford on 4 January 2020. He made his Championship debut for Stoke on 12 July 2020 in a 2–0 victory against Birmingham City in place of the injured Jack Butland. Davies kept his place in the side for the final few matches of the 2019–20 season and impressed as Stoke remained unbeaten to avoid relegation from the Championship.
