Givet has not played for the club since being sent off against Arsenal in February amid speculation over his future.
Now, Kean's revelations seemed to hint at a lack of desire to play from the Frenchman as the Blackburn manager compared his situation to that of David Dunn.
"It was apparent when I spoke to Gael this morning he wasn't anywhere near in the right frame of mind to play today, whether starting or on the bench. I will speak to him again, and if he is in a better frame of mind then great, but he was nowhere near in the right frame of mind today," said the Scot.
"I can't get inside somebody's head, I can only say how much we and his teammates need him. If you look at David Dunn with his hamstring, he is having injections to play.
"We need guys who want to play even if they are injured or their heads are not quite right, but I would rather keep the conversation between myself and Gael private. We need everyone to keep pushing as I believe we need two wins and a draw from our remaining games [to stay up]."
Rovers currently sit 19th in the Premier League table, two points away from safety.