Rumours began swirling during the week that Postecoglou had emerged as Daniel Levy's preferred candidate to take the reigns at Spurs.
The Australian coach did not let the intensifying speculation affect his plans though, with Celtic running out as comfortable 3-1 winners against Inverness to secure a world record-breaking eighth domestic treble.
Kyogo Furuhashi (28), Liel Abada (21) and Jota (24) were on the scoresheet at Hampden Park on Saturday as the Bhoys rarely ever looked in trouble during the match.
The win rounded off an impressive season for Celtic and Postecoglou, marking his fifth major trophy since taking over from Neil Lennon at Parkhead in 2021.
The Celtic boss was keen to bat off questions on the subject of his future after Saturday's final.
"I understand all that, but I am going to be a little bit selfish here, not just for myself but my family and people around me," he told BBC Scotland.
"We have all worked really hard to enjoy this moment.
"If I again start talking about things right now that aren't really important to me and those people around me... I am just enjoying this and will do for as long as I can before people drag me away to talk about other things.
"I am just super proud of them and am amazed at the levels they have created. I think everyone is looking forward to a break."
Reports suggest Tottenham have yet to make an approach for Postecoglou, but now that the Scottish season has ended they are expected to move quickly in the coming days.
He is currently on a 12-month rolling contract in Glasgow and Celtic could yet offer him a bumper new long-term deal in the hope of convincing him to stay in Scotland.
Spurs are also rumoured to be speaking with Brendan Rodgers (50) and Luis Enrique (53) regarding the vacancy in north London.
