Bunn is on a month's loan from Ewood Park and is currently the first-choice goalkeeper while the Blades await the outcome of Paddy Kenny's hearing regarding his failed drugs test.
And with the hearing date put back to beyond the end of Bunn's scheduled stay, 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.
"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."