The treble-winning 57-year-old is believed to be on the verge of becoming the Lilywhites' new manager, and Spurs are hopeful of wrapping up a deal within the coming days.
Daniel Levy turned his attention to Postecoglou after Julian Nagelsmann and Arne Slot both dropped out of the running, while a reunion with Mauricio Pochettino did not come to fruition as the Argentine opted to join Chelsea.
Postecoglou - who is only on a one-year rolling contract with Celtic - is therefore poised to take over at Tottenham after guiding Celtic to their eighth treble in the 2022-23 season, clinching the Scottish Premiership, Scottish FA Cup and Scottish League Cup.
The most recent of those triumphs came in Saturday's Scottish FA Cup final, where the Bhoys defeated Inverness Caledonian Thistle 3-1 at Hampden Park, where Furuhashi scored the game's opening goal.
Having forced a highly successful relationship with Furuhasi at Celtic Park, The Sun claims that Postecoglou would be keen to continue working with the Japanese striker in North London.
Furuhasi left Vissel Kobe in 2021 to follow Postecoglou to Scotland, and the 28-year-old ended a stunning 2022-23 season with 34 goals and five assists from 50 matches in all competitions.
Furuhashi won the Scottish Premiership Golden Boot with 27 strikes in 36 games - two more than second-place Kevin van Veen of Motherwell - although he was not called up to the Japan squad for the 2022 World Cup.
The 5ft 7in forward boasts a total of 54 goals and 10 assists from 83 outings for Celtic since 2021, but his contract with the Scottish champions only runs for another two years.
The report adds that Spurs are willing to pay at least £25m and up to £30m to capture Furuhashi, but they will have to fend off the attention of other English clubs in the coming weeks.
Furuhashi could arrive at Tottenham as a backup or replacement for Harry Kane, whose future remains a hotly-debated topic in the wake of another trophyless season for the Lilywhites.
The 29-year-old will soon enter the final year of his contract with Tottenham and is believed to be on the radar of Manchester United, Real Madrid, Bayern Munich and Paris Saint-Germain.
Spurs chairman Levy is still hopeful of convincing Kane to sign a new contract, but the England captain will be able to leave on a free transfer in the summer of 2024 as things stand.
Kane has already confirmed his desire to break Alan Shearer's all-time Premier League goalscoring record as Man United try to steal a march on their rivals, but Levy is reluctant to sell to a fellow English top-flight club.
Kane netted an impressive 30 goals for Spurs in the 2022-23 Premier League season, but the Lilywhites finished outside the European places in eighth following another turbulent year.