Speaking after his side's 1-1 draw with Stoke, Bruce confirmed his summer-long pursuit of 30-year-old Dawson was close to bearing fruit and hopes to push the transfer through quickly.
But having seen Blackburn reject two bids for striker Rhodes, there is plenty of work still to do if he is to replace Shane Long up front.
"It's up to me to bring two or three in, we're close to one in particular and we hope we can get that over the line," said Bruce.
"Michael Dawson we hope we can get. There's a deal in place in principle anyway. Hopefully, touchwood we'll see what develops.
"I thought I had Jordan Rhodes 24 hours ago and it seems to be further away than ever. We've been negotiating for a week.
"We'll see what this week brings and hopefully bring in the two or three we need to give the squad a hand because we've already played five competitive games."
- Michael Dawson
- Jordan Rhodes