The 19-year-old has been on the Citizens' radar for some time and was linked with a move to the Etihad Stadium during the summer transfer window.
Both Brighton & Hove Albion and Juventus have also been credited with an interest in Barco, but it appears that Man City have moved to the front of the queue for the Argentine's signature.
Pep Guardiola's side allegedly tabled a 'tempting offer' for Barco that exceeded a bid from Brighton earlier this summer, but a switch to the blue side of Manchester failed to materialise for the youngster.
Nevertheless, according to Argentina-based journalist German Garcia Grova, Man City now have an agreement in principle with Barco, and emissaries of the treble winners are set to travel to South America in order to strike a deal with Boca Juniors ahead of a potential move in January.
It is thought that City Football Group director Joan Patsy, a South American-based chief who was a key figure behind the signings of Gabriel Jesus and Julian Alvarez, will be involved in the negotiations for Barco.
Born in Buenos Aires, Barco has attracted interest from across Europe since breaking into the Boca Juniors senior side earlier this year and establishing himself as a first-team regular.
Barco has made 21 first-team appearances for the Argentine giants in all competitions, including 11 league starts and six outings in the Copa Libertadores this year.
After scoring his first senior goal in a Copa Libertadores group-stage win over Monagas SC, the versatile left-flanker – who can play as either a left-back or left winger – will soon endeavour to help Boca Juniors advance to the final of
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