Karanka hopes to use his relationship with Blues boss Jose Mourinho, to whom he was number two at Real Madrid, to improve competition for places within his squad with the addition of the two talented youngsters.
Speaking after watching Boro win 3-0 at Oldham Athletic in the first round of the Capital One Cup, Karanka told reporters: "I hope we will sign the two from Chelsea. The club are working on that, but we have got seven games in 22 days so I'm concentrating on the games.
"We are improving every day and showing we are a good team, but I also want to sign more players. Maybe two or three more signings will be enough for me to have the squad that I want.
"We are not strong enough at the moment. I want players who can't sleep because they are worried that someone else will take their place."
Bamford, a 20-year-old striker, impressed during spells at MK Dons and Derby County last season, for whom he scored a combined 22 goals in 44 games, while Saville, 21, helped Brentford win promotion from League One. Both have been linked with moves to Wolverhampton Wanderers.