Standard Sport understands the 57-year-old is the leading managerial target at Spurs and attempts to appoint him will be ramped up this coming week.
The Glasgow giants beat Inverness to seal a treble on Saturday and Postecoglou remained coy on his future after the victory at Hampden Park.
Appointed in 2021, little was known about the former Yokohama F. Marinos and Australia manager prior to his arrival.
While the decision to replace Neil Lennon with Postecoglou may have been derided in some quarters, Celtic have roared back under his watch after Rangers won the Scottish Premiership for the first time in nine years.
Postecoglou admits he was seen as a “joke” in some circles when walking into the club and believes it “was up to me” to repay supporters’ trust.
Speaking after Celtic’s win, he said: “I have appreciated the support I have had from day one. I was a joke, a lot of people made fun of my appointment.
“The supporters put their collective arms around me. It was up to me to repay their trust.”
