Manchester City Football Club has been working with the entertainment firm Pophouse Entertainment on the venue, close to the Co-op Live arena.
According to council planning documents, the three-storey venue would be open throughout the year and it would create about 150 jobs.
Jessica Koravos, the Chief Executive of Pophouse, said: “Manchester is known for its innovation in music and entertainment and we cannot think of anywhere better to bring one of the world’s most popular and ground breaking theatrical productions”.
“Mamma Mia! The Party has been hugely successful in London and we are excited to be able to expand on that success,” she added.