According to Sky Sports, the Hoops have been tracking the defender since the beginning of the summer and are keen to sign him on a loan deal.
He was invited to Celtic Park to watch the Europa League clash against Jablonec last night.
The Hoops picked up a 3-0 win and Couto is likely to have been impressed with the performance, but he has not decided his next move yet.
Ange Postecoglou needs to add more depth and quality to his defence, and the Brazilian full-back could prove to be a useful option for him.
The 19-year-old is unlikely to get regular game time at City this season, and a move to Parkhead could be ideal for his development.
Regular game time at the Scottish club will help him improve as a footballer, and he could return to City as a better player next summer.
The Scottish giants have also been linked with Aurelio Buta, Josip Juranovic and Ko Itakura in recent weeks.
Couto was on loan at Girona last season, and a move to Celtic would allow him to further his development.
He would get regular game time at a high level, and the Hoops will also be able to provide him with the experience of European football.