First choice Paddy Kenny has opted to team back up with former Blades manager Neil Warnock at QPR after serving his nine-month drugs ban, leaving Blackwell without a senior goalkeeper.
Steve Simonsen, who stepped in on loan from Stoke towards the end of last season, remains one of several options open to Blackwell.
Blackwell said: "Steve did well and we know all about him but it's far from nailed on and there are other quality goalkeepers about such as Jamie Ashdown (free agent) and Marton Fulop (Sunderland), who we are also looking at.
"I would really like to tie a deal up by next week because we had major problems last season by not starting with a regular number one.
"It's very important we get one in and bedded down quickly."