The Lilywhites have been without a permanent head coach since Antonio Conte's departure by mutual consent in March, which came after he had blasted his "selfish" players for throwing away a two-goal lead against Southampton.
Daniel Levy, who is also currently without a sporting director following Fabio Paratici's resignation, has suffered a series of setbacks in his search for a new head coach, with several contenders dropping out of the running.
Julian Nagelsmann and Arne Slot are among previous top candidates to have pulled out of the race, while a possible reunion with Mauricio Pochettino was a non-starter as the Argentine headed for Chelsea.
After conceding defeat in their efforts to prise Slot away from Feyenoord, Tottenham apparently turned their attention to Postecoglou, who has enjoyed a trophy-laden two years at the Celtic helm.
The 57-year-old refused to add fuel to the fire surrounding his future when questioned on Tottenham links last week, but according to The Athletic, an appointment is now close.
The report states that Spurs are hopeful of wrapping up a deal to appoint the Australian within a few days, and Levy is hopeful that Postecoglou can oversee a "culture of change" in North London.
The former Australia manager took over from interim head coach John Kennedy in 2021 after Neil Lennon departed for a second time, but he was only appointed on a one-year rolling contract.
As a result, Spurs will not have to pay compensation to Celtic, which was believed to have been a roadblock in their pursuits of Slot and ex-Bayern Munich head coach Nagelsmann.
Postecoglou boasts a record of 83 wins, 12 draws and 18 defeats from 113 games in charge of the Bhoys, who won a record-extending eighth Scottish domestic treble in the 2022-23 season.
Postecoglou had already masterminded Celtic's runs to glory in the Scottish Premiership and Scottish League Cup prior to Saturday's Scottish Cup final versus Inverness Caledonian Thistle, where a 3-1 win sealed a trio of trophies for the Hoops.
The Athens-born coach also collected a Premiership and League Cup double with Celtic in the 2021-22 season and will take over at Tottenham following Cristian Stellini and Ryan Mason's underwhelming periods in interim charge.
Instead of leaving with Conte, the Italian's assistant Stellini was promoted to the top job after his compatriot left the club, but he was sacked after a humiliating 6-1 loss to Newcastle United, lasting just 29 days in the hotseat.
Mason did not fare much better than Stellini, with Spurs winning just two of their six games under the 31-year-old, who also saw out the 2020-21 season after Jose Mourinho's departure.
Despite relegating Leeds United with a 4-1 win on the final day of the Premier League season, Tottenham could only finish eighth in the table and will be without European football for the first time since the 2009-10 season next year.