The 20-year-old forward is back at Stamford Bridge after splitting the 2013/14 campaign between MK Dons and Derby County.
He proved to be a regular source of goals in the second and third tiers but is not expected to break into the first team set-up at Chelsea any time soon.
Another loan deal has been mooted throughout the summer, with a number of Sky Bet Championship sides expressing an interest in securing his services.
Middlesbrough appear to have edged to the front of the queue, with discussions held with club and player.
No agreement has been confirmed, though, so Karanka is reluctant to speculate on when Boro fans can expect to see Bamford joining their ranks.
He told the club's YouTube channel: "I hope he can be (a Boro player) but it's not in my hands. It's with him.
"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."