The 15-year-old is the latest off the Charlton academy production line - after the emergence of Ademola Lookman, Joe Gomez and Ezri Konsa in recent years - and has already appeared for the Addicks' Under-23s.
It is claimed the League One side have offered Spain-born Sarmiento a five-year deal to ward off suitors, but are looking for £6 million to cash in.
The report adds, however, that any fee would be set by Football League tribunal - although Charlton stand to receive a record amount for a schoolboy player, which was set when Chelsea paid £2.5 million for Exeter City's Ethan Ampadu in 2017.
And it also states that Manchester City have not been put off by the Addicks' valuation of the teenager, who plays as a winger or number 10.
A move to City, the newly-crowned Premier League champions with a state of the art training complex, would appeal to most young footballers.
Could Charlton lose Sarmiento to Manchester City?