Bunn is one a month's loan from Ewood Park and currently finds himself the first choice keeper while Paddy Kenny is out of the game because of a failed drug test.
Kenny must sit through a hearing, but that is not due to start until after Bunn has gone back to Ewood Park, so Blackwell is keen to keep the 24-year-old at Bramall Lane.
"There is an option to keep him. We've got to wait for the outcome (of the hearing), knowing what the outcome will be is proving very tricky. He said.
"We will have to go cap in hand to Big Sam (Allardyce) and hope that Blackburn will let him stay for a bit longer.
"He has shown is a keeper of potential, he is bright about his box and we are delighted to have him. Hopefully if the situation is that we need to have him a bit longer we can."