The club's all-time leading goalscorer has entered the final 12 months of his contract and has shown no intention to pen fresh terms.
Spurs chairman Daniel Levy had been insistent over retaining the services of the England international until it was alleged that owner Joe Lewis had demanded a sale or a renewal of his deal.
That latest development will only have encouraged Bayern Munich, who had have two bids rejected yet remain waiting in the wings for the 29-year-old.
Nevertheless, like he did against West Ham United in Australia last week, Kane is still featuring under Ange Postecoglou as though he will start the season at the Tottenham Hotspur Stadium.
Kane was named in the starting lineup for their second fixture of the summer, an encounter with Lion City Sailors in front of over 25,000 supporters in Singapore.
Their opponents, who sit in third place in the top flight, took a surprise lead during the first half before Kane got himself on the scoresheet with a penalty just before half time.
Postecoglou opted to withdraw Kane at half time, just as he did versus the Hammers, but Spurs proceeded to kick on and register a 5-1 victory.
Richarlison, who endured a disappointing first season in North London, scored twice within seven minutes of his introduction before completing his treble in added-on time.
Giovani Lo Celso scored the other goal in the 73rd minute, backing up his strike versus West Ham, with the Argentine potentially putting himself in line to remain at Spurs rather than move elsewhere.
Spurs are still due to play the likes of Shakhtar Donetsk and Barcelona before the start of the new Premier League campaign, which for them begins with a fixture at Brentford on August 13.
Postecoglou will hope for some clarity over Kane's future by then as he looks to build a settled side after making the switch from Celtic earlier in the summer.