The 20-year-old goalscorer enjoyed successful spells at MK Dons and Derby County last season, but has now returned to Chelsea for the new campaign.
"I hope he can be [a Boro player] but it's not in my hands. It's with him," said Karanka via the club's official YouTube channel.
"He can be a very good player for us because we just have Kike as striker but, at the moment, he's not my player."
Despite an agreement not being reached, the clubs have reportedly already held talks about the youngster's plans for this season.