The 17-year-old centre-back has completed his medical and signed a long-term contract with the Premier League club, the transfer expert reported.
Osborne represents Scotland at Under-19 level and had offers from several clubs across Europe before choosing Chelsea, according to Romano.
The move comes as Chelsea look to strengthen their defensive options for the future after finishing tenth in the Premier League last season, a result that leaves them without European football next term.
Xabi Alonso has taken over as Chelsea manager and recently oversaw the signing of Marco Palestra.
Alonso's arrival is reshaping the pecking order across Chelsea's squad, with midfielder Andrey Santos, 22, now facing an uncertain future, according to the BBC.
Chelsea are reportedly ready to sanction a lucrative sale of the Brazilian, whose stock has risen sharply amid a booming market for central midfielders.
Elliot Anderson's £135million (€135m) move to Manchester City and Tottenham's British record deals for Mateus Fernandes (€98m) and, reportedly, Sandro Tonali (€115m) have driven fees for the position to new heights.
Chelsea remain under scrutiny from UEFA and the Premier League over their financial sustainability, adding further incentive to cash in on Santos.
Santos joined Chelsea from Santos in 2022 for €23million and has already attracted plenty of interest, the BBC reported.
Enzo Fernández's own Chelsea future also remains unresolved, with the midfielder said to want a move away against the club's wishes.
Chelsea's summer business therefore looks set to bring both incomings and lucrative outgoings as Alonso reshapes his midfield and defensive options.