The 19-year-old has established himself as one of South America's top prospects through his performances for the Argentina giants.
A total of 35 appearances have been made in all competitions, chipping in with one goal and four assists, but there is an acknowledgement that his time at Boca Juniors may soon come to an end.
Unless a contract renewal is agreed at some point in the near future, Barco is expected to depart before his contract expires at the end of 2024.
Boca Juniors are aware that a sale in January may make sense for all parties and that is seemingly the direction that they are heading in at this point.
While there have been a wide array of clubs who have been credited with an interest in the youngster, Man City allegedly have the edge over the likes of Liverpool, Brighton & Hove Albion and Chelsea.
City's interest in Barco also comes at a time when Pep Guardiola may be in the market for a new left-back, the likelihood being that Sergio Gomez could move onto pastures new.
Nevertheless, according to The Sun, Guardiola's preference would be for Barco to gain regular football elsewhere ahead of competing for a first-team spot at the Etihad Stadium.
The report claims that the Spaniard would look to Enzo Maresca - who worked on his backroom staff before becoming head coach of Leicester - to help with his development.
Leicester already have a settled squad that is sitting three points clear at the top of the Championship standings as they bid to earn an immediate return to the Premier League.
Just 10 goals have been conceded in 17 games, the injury to Callum Doyle has left the position as arguably a problem area with the versatile James Justin currently holding down the role.
Man City would reportedly pay in excess of £10m in order to win the race for Barco, who has represented his country at Under-20 level.
If City do not complete a deal in January, Guardiola is said to be open to Barco spending 2024-25 at the King Power Stadium, the likelihood being that the Foxes will be in the top flight.