"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."
Saturday's 2-0 home Championship defeat to Brighton was the fifth time in Boro's last six games that they have failed to score.
"I'm saying the same things every week because it's obvious what the problem is," Mowbray added.
"The boys are working hard but nothing is dropping and we can't score.
"These defeats against Hull and Brighton, were they any better than us? No. But we are not winning football matches.
"I know the fans are angry and upset. They want us in the Premier League. So do I and so does the chairman.
"We're going backwards at the moment and ultimately the chairman will make the big decisions, but I won't quit. Let's see what the summer brings."