The 18-year-old is due in Glasgow for a medical over the weekend.
The England under-19 international made the move from Fulham to City in the summer for a fee thought to be in the region of £12m.
"He is a very good player and I am sure that Patrick will be a very important player here in the future," said City boss Manuel Pellegrini.
"I thought it was important that he trained with our first team for the first half of the season.
"But to play more games for him at Celtic will be better than to stay here and just play some games. It will be a very important option for him."
Roberts has made three appearances as a substitute for City this season, having played 22 first-team games for Fulham.
Celtic took striker John Guidetti and defender Jason Denayer on loan from Manchester City last season and signed Dedryck Boyata from them this summer.
Celtic have agreed an 18-month loan deal for Manchester City midfielder Patrick Roberts.