Postecoglou faces having to win the Europa League to save his job after Spurs slumped to a 15th Premier League defeat of the season at Fulham on Sunday.
Sources have told talkSPORT that Bournemouth boss Andoni Iraola would be Spurs chairman Daniel Levy's top target.
Marco Silva and Thomas Frank of Fulham and Brentford respectively are other potential targets, although both have hefty release clauses in their contracts.
Postecoglou is only in his second season at Spurs, although it's been a campaign in which he's dealt with various injury issues.
However, should results continue to fall below expectations, then the club hierarchy could find themselves deciding to part.
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Iraola is unsurprisingly high on their list of potential replacements amid his ongoing success at Bournemouth.
Although they sit tenth in the Premier League, they are still fighting for possible European qualification.
Achieving such a prospect would be a huge milestone for a club that were playing in the Championship as recent as 2022.
Spurs fan tension was on show following Sunday's defeat 2-0 to Fulham.
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Postecoglou was involved in a heated confrontation with a supporter and told them to 'behave'.
It's not the first time this season that the Australian has faced vocal criticism from fans at matches.
He was booed went making his way to the away end after their 1-0 defeat at Bournemouth in December.
Spurs loanee Mathys Tel also urged fans to get behind the team after the defeat at Craven Cottage.
A top half finish this season has become increasingly unlikely for the north London side given they've won just ten league matches.
FA Cup and Carabao Cup success is also no longer possible, although they could still win the Europa League.
Spurs face Eintracht Frankfurt in the quarter-finals and should they advance to the semi-finals, they will face Bodo/Glimt or Lazio.
The March international break will likely be welcomed by the team and fans ahead of the crucial season run-in.