Mowbray has seen his side record just two victories in 2013, leaving the club eight points off the Championship playoffs, and the former West Bromwich Albion boss has revealed that he has no intention of resigning at the Riverside Stadium.
The 49-year-old told BBC Tees: "I'm a fighter, I'm not going to walk away from anything.
"I'll never give in. You'd have to ask the chairman about my future. If he feels someone can come in and do a better job, that's his prerogative."
Middlesbrough must beat Nottingham Forest on Tuesday night if they are to retain any realistic hope of reaching the playoffs.