After reportedly failing with two bids for the England captain - who is out of contract in 2024 - Bayern are believed to have lodged a final take-it-or-leave-it offer worth £86m.
The German champions supposedly set chairman Daniel Levy a deadline of Saturday morning to respond to their proposal, as they are keen to move onto other targets if their advances continue to be rebuffed.
However, Levy ignored Bayern's ultimatum, and Kane captained Tottenham to a five-star thumping of Shakhtar, during which he scored four goals in a scintillating individual performance.
Postecoglou was unsurprisingly quizzed on Kane's future in his post-game press conference, but the Spurs boss reacted indifferently when asked about Bayern's approach to their pursuit, having apparently irked some within the club with their public declarations of interest.
"I don't have a view either way mate. That's up to Bayern to decide. If that's the way they want to conduct things, it's not up to me to judge that. It doesn't affect me. I don't sit there on a daily basis and worry about what other clubs are doing," Postecoglou said.
"I think what is factual right now is that he is a contracted player of our football club. I certainly wouldn't talk about contracted players at other football clubs but I'm not at Bayern mate so they can go the way they want.
"From my perspective I don't know what deadlines means. We know there's one deadline and that's the end of the transfer window. Beyond that, I'm in open dialogue with Harry and with the club on a regular basis.
"I've said to them that if the status quo changes then inform me. I don't need to know about the comings and goings in between. I just want to deal with what's in front of me because I've got to still, irrespective of the outcome of that, I'm building a team here.
"I can't wait [around] for a decision either way to get going. We don't have the time or the luxury to do that. I'm working with what's in front of me. You saw today that Harry certainly is invested in what we're doing and we'll keep on doing that unless something changes."
Dane Scarlett was also on target for Tottenham in Sunday's charity friendly, where Wolfsburg defender Micky van de Ven was in attendance ahead of his £43m transfer to North London.
The Dutchman's signing is expected to be announced in the coming days, and Postecoglou affirmed that Spurs are in need of different profiles at the back, having also captured 18-year-old Ashley Phillips from Blackburn Rovers in a £2m deal.
"It hasn't been done yet so it's not right for me to talk about but there's no doubt we need some bolstering in that central defensive area. We're looking for players that can play the kind of football we want," Postecoglou said of Van de Ven.
"We need defenders who are going to be able to work in an aggressive formation, be able to defend really well one vs one and the space in behind. Hopefully we'll get one or two done in the next couple of days to bolster us in that area."
Tottenham take on Barcelona in their final pre-season friendly on Tuesday with the Joan Gamper Trophy on the line, before opening their Premier League season with a London derby away to Brentford on August 13.