The parent company that owns Manchester City, City Football Group, also owns Melbourne City and helped to facilitate the transfer.
Arzani flew into the UK on Wednesday to finalise the terms of his contract with Pep Guardiola's team.
"In a short period of time, Daniel has developed into one of Australia’s brightest young stars and has demonstrated the drive and raw talent required for success at the highest level," said City Football Group's managing director of global football, Brian Marwood.
"This pathway was first forged by Aaron Mooy just two years ago, and we are all delighted to see another Australian-grown talent make this move.
"The combination of City Football Group’s global football structure and the world-leading City Football Academy facilities create an environment in which this pathway can become well-trodden."
Arzani was the youngest player at this summer's World Cup and the youngest player to play at the tournament in the history of Australia. He impressed as a substitute in all three of the Socceroos' group matches.
The 19-year-old, who was born in Iran, earned a host of admirers in his breakthrough season in the A-League and was given the league's Young Footballer of the Year award for 2017/18. He also won the Harry Kewell medal for the best U23 player in Australia.
It is expected that Arzani will join Celtic on loan for the duration of the 2018/19 season, with City referencing a "career development path" in their official statement.