Bruce's men are third-bottom in the top flight, having claimed just four league wins all season, and were soundly beaten 3-0 at home by a mid-table Newcastle United side last Saturday.
There has been speculation that Bruce - who got a new assistant in the form of Mike Phelan on Thursday - could be living on borrowed time at the KC Stadium.
However, speaking ahead of Saturday's visit to defending champions Manchester City, Bruce insisted his relationship with chairman Assem Allam has not changed.
"We are in the bottom three so there is a problem," Bruce said.
"I can't control what a newspaper says or what the speculation is. I am not going to take anything for granted but I would think the chairman, given the relationship we have, would tell me if I was in trouble.
"He has backed me with a new assistant manager, he's backed me with a new striker [Dame N'Doye].
"We have worked very hard over the last two-and-a-half-years and I am not going to give it up without a fight.
"We have 15 games to stay in this division. We are always going to be at the bottom end of the division, you'd expect Hull to be in the bottom six or eight. That's the way it is."
Bruce will not be able to call on deadline day signing N'Doye at the Etihad Stadium as the Senegal striker is still awaiting international clearance.
However, Bruce feels the Yorkshire club should have plenty of hope going into the daunting away trip.
"Middlesbrough and Burnley have got results [at the Etihad] in the recent past," he added.
"What we have to do is get over the disappointment of the last few weeks. If we make mistakes against City then we've no chance but if we roll our sleeves up then we've got a chance."
