On Thursday, Tottenham drew 1-1 with Rangers in the Europa League, with Dejan Kulusevski netting with 15 minutes to go to pull his side back into the clash.
Pressure is mounting on the head coach who was booed after Spurs' 1-0 loss to Bournemouth, with his side now not winning in their last five games.
The Tottenham manager's first season in the Premier League was one of promise after a fifth place finish, but they now sit in eleventh.
Postecoglou's mission is to end Spurs' 16 year trophy drought - and that hope was boosted when they beat Manchester City 2-1 in the Carabao Cup.
Cristian Romero alluded that there had been a lack of investment from the Spurs board compared to their rivals - but Postecoglou still isn't buying that argument.
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Asked if that impacts their chances of winning trophies before his side's trip to Southampton, he said: "I'm sick of that. That's an excuse for me. That's why, even with Cristian Romero's comments, I just think that's an excuse.
"At the moment, I'm here, I take responsibility, these players, we're here.
"If we think some mythical thing exists in this club that prevents success then change it. What's the point otherwise? Don't come here. I just don't buy into that.
"At the moment, we're in a difficult situation because of where we are squad-wise and you either embrace this challenge. If you want excuses, there's a million of them to why we can't be successful.
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"If that's what your clutch is, particularly in tough moments, what's the point? Move on then. Go somewhere else."
One player that came under fire in Spurs' draw at Ibrox was Werner.
The German lost possession 16 times, won none of his five duels and only completed 69 per cent of his passes.
His performance saw him substituted at half time for goal scorer Kulusevski - and his manager didn't hold back on his assessment of him post-match.
After the game, Postecoglou said on Werner: "When you've got 18-year-olds [in the squad], it [that performance] is not acceptable to me. I said that to Timo, he's a German international.
"We need everybody, including him, to be contributing because we don't have the depth to leave people out if they perform poorly. I expect a level of performance from some of the senior guys, and tonight wasn't that."
When asked whether he regrets his comments towards the ex-Chelsea player at the Friday press conference he said: "No, it wasn't criticism, it was assessment. If we've played poorly, we've played poorly. With Timo, it wasn't about whether he'd played well or not, we had a really difficult game.
"I'm asking 18-year-olds to do massive jobs and you've got senior players - he's a senior player, he's an international, won the Champions League - that there's a level of application and performance you need to rise to, to help the team. He didn't reach that.
"Me taking him off at half-time was a statement enough. The same way I won't hide behind team performances, it's not different to players or anyone else."
He was also questioned on how Werner may respond to his criticism and he did not sway from his original opinion.
"I've got no time for that sort of stuff anymore," said the Australian. "We're in a fight here, collectively we're in a fight. I'm not going to go around worrying about people's bruised egos.
"We want to achieve things, we want to be successful. We're down to the bare bones in players, if there's somebody in that dressing room who's fit, able to contribute and feels like they need something extra in this moment, he's probably not the right type.
He continued: "We had some last night that understood the situation that we're in. We've got 15 fit players, in some positions we've only got two players.
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"I'm not going to go around trying to get extra out of people if they don't give extra right now. That gives me an indication of where they're at."
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RB Leipzig loanee Werner has one goal and three assists this season in 19 matches in all competitions.