Arzani has just finished representing Australia at the World Cup, with the winger making three substitute outings during the tournament in Russia.
The 19-year-old is regarded as his nation's next star and according to the Manchester Evening News, City are monitoring the player's development.
Arzani represents Melbourne City - the sister team of the Premier League champions - and the English giants are considering making an official bid for the attacker.
City took a similar approach with Arzani's compatriot Aaron Mooy in 2016, with a permanent deal being done before the player was loaned out to Huddersfield Town to gain experience of British football.
Arzani has just a year remaining on his current contract, increasing the chances of a switch to a European club this summer.