The Sun reports that the trio are weighing up a £20m ($36m) bid to land the 21-year-old in the upcoming transfer window, with the player showing his flair by scoring four goals in 18 appearances this campaign. United are in the market for a midfielder after being overrun in the middle of the park by Manchester City in their recent 2-1 loss to their arch rivals at Old Trafford.
Amiri was part of the German under-21 side which won the Euros in the summer, beating England in the semi-final and Spain in the final. Amiri has been a regular feature for the Germany Under-21 squad and recently scored in a 7-0 thrashing of Azerbaijan. The midfielder is noted for his versatility, and is also capable of playing on the left wing if required.
All three clubs are currently looking into the market for potential deals in the upcoming window, with a title challenge looking difficult, given the Citizens' current form.
He would come in as a potential replacement for Henrikh Mkhitaryan at United, who has fallen out of favour with Jose Mourinho.
The Armenian international started the season on a bright note, registering five assists in the first three Premier League fixtures. Since the end of August, he has managed to score twice and has failed to register an assist in all competitions.
Meanwhile, there is talk of a potential departure for Jack Wilshere from the Emirates, which would open up another spot in midfield, allowing Arsene Wenger to bring in someone in January.
The 25-year-old came up through the ranks of the north London club's youth system before breaking into their first squad in 2008. The midfielder was backed to have a bright future at the club, but injury setbacks have stopped his progress at Arsenal.