"He has been up here today so we hope for that to complete over the next few days," said Celtic manager Brendan Rodgers.
"He hasn't done anything since his last game with Australia a little while back so he is going to have to come in and adjust and get up to speed before we can even consider him for a game".
The Iran-born 19-year-old was the youngest ever member of an Australia squad for a World Cup finals and played in all three of their group matches in Russia.
Arzani -- who put down his talent to playing street football in his birthplace of Khorramabad -- follows another Australian Aaron Mooy, who moved from Melbourne City to Manchester City in 2016, and was immediately loaned to Huddersfield Town.
"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," Brian Marwood, the managing director of global football for City Group, told the club website.
"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."
Arzani -- who was also the youngest player at the World Cup finals -- joined Melbourne in 2016.
He enjoyed a breakout season in the 2017-18 A-League campaign as his exciting attacking play won the league's Young Footballer of the Year award.