Morton is a versatile player across the midfield. He joined the academy at 7 years of age after arriving from Greenleas in Wallasey. Morton made his first-team debut as a substitute in a 3-0 win at Norwich in the third round of the Carabao Cup on 21 September 2021. He followed this up with two Premier League appearances before Christmas, as well as playing all ninety minutes of a Champions League game with Porto.

To continue his development Morton was loaned to Blackburn Rovers for 2022/23, featuring forty times in The Championship, especially impressing when playing in a deep lying midfield role. He signed a new long term contact with Liverpool in January 2023. Morton joined Hull City on loan for the remainder of the 2023/24 season on 1 September 2023. He had an impressive spell there, scoring three times in 39 appearances.

After two years developing on loan at Blackburn and Hull City, Morton returned to Liverpool in the summer of 2024 hoping to stake a claim for a place in the senior squad. Klopp opted to keep the Wirral-born midfielder at Anfield rather than send him out again, giving him a chance to integrate into the first team. Morton featured in Liverpool’s pre-season schedule and remained part of the first-team group throughout the campaign. He made his first competitive appearance of the season in the EFL Cup and added further appearances in the domestic cups and as a substitute in the Champions League. Morton's opportunities were limited due to strong competition in midfield and he only managed to make five first-team appearances across all competitions in the 2024/25 campaign. He showed glimpses of the calm, composed style that had earned him praise during his earlier loan spells—particularly his positional sense and tidy passing under pressure. Morton was unfortunately sidelined in early February 2025 due to a shoulder injury that required surgery and was effectively absent for approximately three months.

Competition for places in Liverpool's midfield only increased in the summer of 2025 and Morton was transferred to Lyon in French Ligue 1 prior to the 2025/26 season for £8.7 million plus £4.3 million in add-ons. Liverpool have a 20 per cent sell-on clause of any future move.