The Championship club are in active talks with West Ham about the 24-year-old, who made just 10 appearances for the Hammers last season.
When asked about the interest in Byram, Karanka said: "He is one on the list. He is a West Ham player and we are not in a hurry. We have another few days to try to buy someone and another two or three weeks to try and bring someone in on loan.
"We have been working since May to try to build a good squad and we will bring in the right one."