The 19-year-old is regarded as one of the hottest prospects in Australian football and became the youngest player ever to represent his country at a World Cup when he featured in Russia this summer.
Arzani is thought to have penned a long-term deal with City but is now expected to be loaned out to Celtic for two years in order to gain first-team experience.
Brian Marwood, MD of global football at the City Football Group, said: "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.
"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."
City have made just one first-team signing so far this summer, bringing in Riyad Mahrez from Leicester City in a £60m deal.