The England international has fallen down the pecking order at White Hart Lane since Mauricio Pochettino's arrival from Southampton and now looks set to move to the Tigers
The 30-year-old, who has won four England caps, appears to be surplus to requirements at White Hart Lane and Bruce is keen to bring him to the KC Stadium.
"Managers are always going to say we are thin on the ground," said Bruce after Sunday's 1-1 Premier League draw with Stoke City.
"It's my job to try and bring players in, but we're close to one in particular that we hope we can get over the line, although I thought I had Jordan Rhodes 24 hours ago and that now seems to be further away than it's ever been after being negotiated for a week."
When asked about Dawson, though, Bruce added: "I'm hopeful that can be done, I believe there is a deal in principle.
"But we'll see what this week brings and whether we can bring in the two or three people we need to give the squad a hand because we've already played five competitive games."
Hull had two bids for Blackburn Rovers striker Rhodes "respectfully and categorically rejected", the Championship club confirmed on Saturday.