Bruce openly declared his interest in the Sunderland striker earlier this week and has wasted little time in opening formal negotiations between the two clubs.
Talks are now at an advanced stage and the 27-year-old striker looks likely to join up with a Hull side readying themselves for life back in the Premier League.
"We're very close and we're hopeful we can get it done before the weekend," Bruce told The Hull Daily Mail.
"It looks as though there's a deal to be done with Sunderland and that's now in the hands of our chief executive.
"All the talks have been encouraging but we'll have to wait and see now. We hope we can get something done soon.
"As I've admitted, he's a player I like and it would be a big boost to our squad to have someone like Danny join."
Sunderland signed Graham from Swansea in January, but he has drawn a blank in front of goal in his 13 appearances for the club to date.