The Australian coach, 57, has emerged as the bookmakers’ favourite for the Spurs job this week after Feyenoord boss Arne Slot joined Julian Nagelsmann, Thomas Tuchel and Mauricio Pochettino in being ruled out of the hunt.
Speaking ahead of next weekend’s Scottish Cup final date with Inverness, Postecoglou dismissed the notion that winning a third trophy of the season would prompt him to finding a new challenge.
“That doesn't really change,” he said. “There is always stuff to achieve. As a football club we have the responsibility to deliver success and play football a certain way.
“Just because we won the double last year doesn't mean we came into this year thinking, okay, we don't have to be as strong. We had to be stronger. The challenge is always there, irrespective of what we have achieved so far.
“And if ever you kind of lose that focus then you probably find yourself out of a job because that's the demands this football club places on the people who represent it.
“We have a chance to win a cup final and make it a special year and next year we all start on zero again and the opportunity is there to try and improve and be stronger for what's ahead.”
Postecoglou was also in the frame for the Chelsea job before Pochettino moved ahead as the leading candidate.
On the latest links to Tottenham, he said: “I'm aware of it because everyone keeps asking me about it.
“It's happened in the past, I have been pretty clear it's just not the way I think and work. My priority right now is just making sure we have a special day on Saturday and we have a cup final coming up in a week's time.
“I'm not going to sit there and get distracted by things that are kind of fascinating for other people rather than myself. That's where my focus is and where it will continue to be. We have had a fantastic season and we have an opportunity to make it a really special one.”