Osman said: "It's a great battle with Liverpool, the fact that we are both in the same city obviously gives it that extra edge.
"Whether we were both fighting for fourth or 10th, there would always be a bite about it because we want to be above Liverpool in the league. As long as there is a chance we will keep fighting, even if it means going right to the last minute of the final game. We will not give up without a real fight now.
"And it could well go to that last match for both of us. We have big games and probably a harder run-in than Liverpool with games against Chelsea and Arsenal. It would be great if we went into the last game with the race for fourth still not decided. It will say a lot about us and how we have gone about the final part of the season. And I believe it could well go right down to that last match for both of us."
Of his own performance against Derby, Osman said: "I'm pleased, you always are when you score. But it was hard. Derby looked to be enjoying their football, they were relaxed and that made it extra difficult.
"We had to dig in. It was pretty ugly stuff in that first half, but we improved and got the result we needed.
"We have put the pressure back onto Liverpool. While we have the chance to catch them we will go for it. We will keep fighting and see where that takes us."
Osman added: "I only scored three last season, so I have doubled that and more. But there is always more to come from me, I am never satisfied and look for faults in my own game.
"I try to improve, that is part of being a professional. Mind you, it is a little difficult being asked to play various positions, like me.
"I feel I am adaptable but you have to learn the requirements of each position you are asked to play, whether it's wide, central or in the 'hole' behind the striker.
"I enjoy it more in the centre of midfield, and when I did get myself into that position I was able to get behind their midfield with late runs and that is where my goal came from."