The 18-year-old becomes Ronny Deila's second signing of the winter transfer window after Erik Sviatchenko joined The Bhoys a couple of weeks ago from Danish side, Midtjylland.
Roberts is highly rated at The Citizens having been bought from Fulham for £11 million last year.
The right winger played 22 times in all competitions for Fulham between 2014 and 2015 without scoring. Roberts only started two games in the league for The Cottagers with the rest of his appearances coming from the bench. At Manchester City, Roberts has made three substitute showings.
News of the possibility of Roberts' transfer to Celtic has been common knowledge among fans for a few weeks but the England u18/u19 international hadn't yet put pen to paper.
Roberts said he would bring "pace and skill" to Celtic and remarked that his new club is known around the world.
The latest signing from Man City is Celtic's third in recent times from the sky Blues following the loan deals of Jason Denayer and John Guidetti.
Celtic need refreshment after cup exit
Celtic were defeated in the Scottish League Cup semi-final against Ross County on Sunday afternoon, losing by three goals to one. The defeat destroys any idea the domestic treble, something which hasn't been done at Celtic for 15 years.
Roberts was in attendance at Hampden Park for the match and trained with his new team mates on Saturday and on Monday.
The only other possible transfer to Celtic before the end of the window is former Blackburn Rovers attacker, Colin Kazim-Richards. The Turkish international is currently playing with Dutch giants Feyenoord but has fallen out of favour following a threat on a journalist. The deal would be another loan and the London born player has been in Glasgow in the past couple of days.