The Tigers have been tracking the 30-year-old centre-back for much of the summer, but have been unable to tie up a deal as of yet.
But with Spurs closing in on Federico Fazio, negotiations are understood to have kicked on for the experienced defender.
Dawson himself has shown a willingness to stay at White Hart Lane in the past, but new chief Mauricio Pochettino is believed to see him as surplus to requirements.
Speaking to talkSPORT after his side’s 1-1 draw with Stoke, Bruce said that Dawson’s move is close:
“[A deal for Michael Dawson] is edging towards where we want it to be. It’s been going on a long time but he’s worth the wait.
“He’s a really terrific player who we would love to bring into the squad if we can.”
Hull are also thought to be tracking Blackburn striker Jordan Rhodes, and Bruce says that he’s hoping to do more business before the transfer window closes:
“We’d still like to sign two or three players before Monday, September 1 but then again I’m probably like every manager. I’m never satisfied.
“Another two or three additions would be terrific especially with the demands on our players [through competing in the Europa League].”