The 19-year-old was on loan at Girona last season and was quite impressive for the Spanish second-tier side.
According to Sky Sports, Celtic are looking to bring the Brazilian to Parkhead on a season-long loan.
He is highly rated at City and needs to play regularly to continue his development.
The defender played 22 games for Girona last season and will be hoping to build on that if he moves to Celtic.
Ange Postecoglou needs to improve his defensive options, and Celtic have been linked with several players in recent weeks.
The Hoops have already signed Carl Starfelt, and they are thought to be looking at full-backs now.
A move to Celtic would be a major step up in Couto’s career as he would get to play at a higher level in Scotland.
Celtic may also be able to offer him European football next season, which would be appealing to the youngster.
It seems unlikely that City will object to the move, and Celtic will be keen to get the deal over the line over the next few weeks.