Jutkiewicz did not feature in any of Blues' pre-season fixtures or last weekend's Carabao Cup defeat to Cambridge United because of a quad injury.
"Jukey is doing well and has been training with the team but I don't know if he is going to be able to play from the beginning on Saturday," said Karanka.
"I will have a chat with him and the medical staff, but I will not take any risks if it meant losing Jukey for a while.
"If he is fit, he will play and if not, he won't."
With Jonathan Leko still not ready to resume playing duties as he nears the end of his recovery period from an ACL injury, Blues have been light in the striking department, although Karanka is still aiming to bring in another forward before the closure of the transfer window.
Meanwhile, Gary Gardner (knee) is closing in on a return but is unlikely to figure against the Bees.
"Gary Gardner has also been training with us today and he will be tomorrow, but I don't think he will be involved," continued Karanka.
"But if I have a chat with him tomorrow and he feels confident, he is an important and experienced player for us, so we will see.”
Spanish winger Ivan Sanchez could also make Saturday's matchday squad for the first time after training all week and Jeremie Bela is available after recovering from the slight shoulder injury he sustained against Cambridge
Elsewhere in the squad Josh McEachran (ACL) remains sidelined.