Club career
Møller Kühl joined Hillerød Fodbold at five years old, and remained with the club for nine years. She joined FC Nordsjælland in 2018. She made her debut against FC Thy-Thisted Q, and she has since been a regular part of the team's starting line-up. In July 2020, she was named Cup Fighter of the Year when FC Nordsjælland won the 2019–20 Danish Cup. After the match, she was highly praised by several media outlets for her good performance and was also compared to the Danish national team star Pernille Harder. The same season she also helped win bronze for the team, in the club's first season in the league.
Arsenal announced the signing of Møller Kühl on 7 January 2023. On 26 January, she made her Arsenal debut, coming off the bench in the 67th minute in the FA Women's League Cup quarterfinal game against Aston Villa.
Arsenal announced her loan to Everton on 13 January 2024 until the end of the 2023–24 season. She made her debut on the same day in the 3–0 victory over Aston Villa in the FA Cup, coming on as a substitute for Karoline Olesen in the 70th minute. She scored her first goal for the club against Nottingham Forest in the FA Cup, opening scoring in the 7–1 victory and assisting a goal.
On 18 January 2025, Møller Kühl was announced at AS Roma on a two-year contract, with the option of a further year.

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