Sources have told ESPN that Postecoglou is now emerging as a strong contender with the club set to hold talks after Saturday's Scottish Cup Final.
Celtic face Inverness Caledonian Thistle and a victory would mean Postecoglou would deliver the club's eighth domestic treble in its history.
The 57-year-old has won five of the six trophies available since joining the club from Japanese side Yokohama F. Marinos.
Postecoglou, who was born in Greece but grew up in Australia where he later managed the Socceroos for four years, is on a 12-month rolling contract and would therefore be free to move this summer without Celtic owed any compensation, although they are expected to try and ward off Tottenham's interest by offering him a longer deal on better terms.
However, Spurs are keen to hold talks with Postecoglou as chairman Daniel Levy comes under mounting pressure to bring some clarity to the club after sacking Antonio Conte on March 27.
Conte's assistant Cristian Stellini was placed in interim charge but lasted just two games, a home defeat to Bournemouth and a 6-1 thrashing at Newcastle United, before he was replaced by Spurs coach Ryan Mason until the end of the season.
Mason has publicly stated that he wants the job on a permanent basis but Spurs are looking for a more experienced candidate than the 31-year-old.
Luis Enrique and Mainz's Bo Svensson are two names to have been discussed but the club continues to distance themselves from any speculation linking them with former Bayern Munich boss Julian Nagelsmann. Levy is also an admirer of Brendan Rodgers, who left Leicester City by mutual consent on April 2.