Amorim is believed to have been seen as City’s top choice to succeed Pep Guardiola should he decide to leave when his contract ends this summer.
Hugo Viana, City’s incoming sporting director who is close with Amorim both professionally and personally, will finish the season at Sporting.
However, the report adds that United have attempted to persuade Amorim that City would lose their dominance if Guardiola leaves next year.
Since Guardiola took charge in 2016, City have won six Premier League titles, two FA Cups, four League Cups, the Champions League, and the Club World Cup.
Omar Berrada, United’s chief executive who joined from the Etihad Stadium earlier this year, has been reunited at Old Trafford with technical director Jason Wilcox, formerly a coach and academy director at City.
Berrada also facilitated the appointment of chief communications officer Toby Craig, whom he knew from City.
City legends Kevin De Bruyne and Ilkay Gundogan’s contracts expire next year, while stalwarts Ederson, Kyle Walker, John Stones, and Bernardo Silva are entering the final two years of their deals.
Earlier this year, Amorim was considered for the West Ham United position after David Moyes’ exit, but the club chose to go with former Wolves and Sevilla coach Julen Lopetegui.
The 39-year-old has been at the helm at Sporting since 2020, leading them to the Portuguese title in his first season to break a 19-year drought.
The tactician, who favours a 3-4-3 system, would command an £8.3million compensation fee, a sum that the United power brokers find reasonable, given their concerns about being limited to free agents.