The 57-year-old Australian has agreed personal terms and Spurs are thrashing out compensation with a Celtic over the final year of his contract.
It means - barring any late hitches - he will replace Antonio Conte as Tottenham's next permanent boss, with Cristian Stellini and Ryan Mason taking interim charge after Conte's departure in March.
The Hoops completed a domestic treble on Saturday when they won the Scottish Cup with victory over Inverness Caledonian Thistle.
Tottenham have since ramped up their efforts to prise Postecoglou away from Parkhead.
Trusted assistant John Kennedy is also set to follow Postecoglou from Celtic to Tottenham.
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Postecoglou, who will become the first Australian to manage in the Premier League, has enjoyed a fine spell at Celtic since taking over in 2021.
In his first year in charge, he led the Bhoys back to glory in the Scottish Premiership as they reclaimed the title they had lost to Rangers.
This season, Celtic successfully defended the league while also adding both domestic cup competitions to their trophy cabinet.
It was the first post Postecoglou has taken in the Northern Hemisphere having enjoyed plenty of success in Australia and Japan.
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His Yokohama F. Marinos side won the J1 League in 2019 while he has also won league titles with South Melbourne and Brisbane Roar.
But what may have attracted Tottenham has been the attacking brand of football on show under his management which has combined speed and possession.
It has made his Celtic side almost unbeatable in Scotland while imports such as Kyogo Furuhashi, Reo Hatate and Oh Hyeon-gyu have proved successes.
After winning the Scottish Cup on Saturday, Postecoglou was asked about his future but he did not want to get drawn into questions about that.
He said: "I anticipate enjoying this moment for the next 24-48 hours, as long as I can, before someone drags me away and takes my attention away from enjoying something that's been hard-earned."
But Tottenham have wasted no time with a formal announcement expected in the coming days once they have agreed a compensation package with Celtic.
Other managerial candidates that were linked to Spurs included Julian Nagelsmann, Xabi Alonso, Arne Slot and Luis Enrique.