Spurs have won just once in their past seven games in all competitions –- a spectacular 4-0 rout of English champions Manchester City –- to tumble to 11th in the Premier League and ninth in the Europa League.
A 4-3 defeat to London rivals Chelsea on Sunday encapsulated the Jekyll and Hyde nature of the Australian's side in 90 minutes.
Tottenham raced into a 2-0 lead, but gung-ho Postecoglou refused to sit on that advantage and the in-form Blues hit back in a thrilling encounter that swung from end to end.
Postecoglou bemoaned his team's indiscipline after conceding two penalties, converted coolly by Cole Palmer, but there has been little sign that they are learning from their mistakes 18 months into his reign.
Last season at home to Chelsea it was two red cards that cost Postecoglou's men in a 4-1 defeat.
Including that result they have won just 18 of their past 43 Premier League games and signs of the Spurs supporters running out of patience were evident in an angry confrontation between Postecoglou and fans in the away end after last week's 1-0 defeat at Bournemouth.
Injuries have not helped the former Celtic manager, who beat Rangers to the Scottish Premiership title in each of his two seasons in charge in Glasgow.
Goalkeeper Guglielmo Vicario is set to be out for months with a broken ankle, while Postecoglou could have just one fit centre-back available after Cristian Romero and Micky van de Ven had to be replaced against Chelsea.
But playing with the intensity Postecoglou demands runs the risk of injuries mounting, particularly during a run of 12 games in 43 days.
"My role now is to focus on the things I can control and keep preparing the team the best we can to keep progressing as a football team and turn our season around," said Postecoglou.
"I think they are desperate to turn our season around and are really disappointed how it has gone. When you're in that mood you kind of lose that composure and discipline you need."
The visit to Ibrox kicks off a huge week that could define Postecoglou's reign.
Spurs visit bottom-of-the-table Southampton on Sunday before a League Cup quarter-final against Manchester United.
The club are without a trophy in 16 years and Postecoglou boldly vowed earlier in the campaign to end that wait for silverware this season.
Man Utd face Plzen
Rangers have been inconsistent themselves this season, disappointing in the Scottish Premiership but impressive in Europe.
Trailing Celtic by 11 points domestically they sit above Tottenham -- in eighth spot in the Europa League after taking 10 points from their opening five games.
Only the top eight progress directly to the last 16, with a top-24 finish enough to make the playoff round.
Manchester United are 12th and also need to ride out a storm when they visit Viktoria Plzen.
Early positivity around Ruben Amorim's appointment as boss has dissipated after back-to-back defeats to Arsenal and Nottingham Forest.
Off the field, the club lurch from crisis to crisis, with sporting director Dan Ashworth departing on Sunday after just five months in the role.
Lazio, Athletic Bilbao and Eintracht Frankfurt lead the way on 13 points, but all face tough away trips this week.
Lazio travel to sixth-placed Ajax, Frankfurt take on Lyon, while Athletic visit Fenerbahce on Wednesday.
