Both clubs had an option to end the 19-year-old midfielder's deal in January, but the Teessiders have no intention of doing so and are confident the Barclays Premier League side are happy for him to remain at the Riverside Stadium.
Mowbray, who is in regular touch with the Blues' assistant first-team coach, Eddie Newton, about the teenager, told Boro's official website, www.mfc.co.uk: "Eddie Newton was at the training ground on Friday and at the game on Saturday.
"He's not Rafa Benitez, of course, but he's telling me the club policy is not to bring on loan players back.
"He told me they have no intention of bringing him back. Josh is doing great for us and we obviously hope he can stay - but he's Chelsea's player."