Amorim has failed to set the world alight since replacement Erik ten Hag last November, leading Man United to a 15th place finish last season, their worst in the Premier League era.
A mixed start to the 2025-26 season has done little to secure his position, with the League Cup defeat to Grimsby Town a particular lowlight.
Toppmoller, 44, who has been with Frankfurt since the summer of 2023, has now revealed his ambition to one day take charge at Old Trafford.
Speaking on the Phrasenmaeher podcast, he said: "Basically it's a dream to be at Frankfurt. It's simply a fantastic club. Here, as a teenager, I witnessed for the first time that my father was a Bundesliga coach.
“That naturally creates an emotional bond with the club. Thinking about abroad, I often played computer games as a child, always with one team - Manchester United.
“That would definitely nice some day because it's an extraordinary club. When I was a teenager, Manchester United were along with Real Madrid the biggest club of all."