After seeing multiple offers rejected by Tottenham this summer, Bayern have now agreed a deal worth in excess of €100 million (£86.4m) for the England captain.
But despite giving encouragement to Bayern earlier in the transfer window, Kane is now having reservations over a move to the Bundesliga champions.
According to Bild, an unknown club has bettered Bayern’s offer with a deal to join on a free transfer next year when Kane’s contract expires.
The report claims that the proposal has come ‘at short notice’, while Manchester United, Chelsea and Paris Saint-Germain have all retained their interest in Kane.
Bayern have offered Kane a contract worth around £400,000 a week.
The 30-year-old has also met with Thomas Tuchel and Bayern’s hierarchy, who have been attempting him to make the move to Germany throughout the summer.
Kane, however, is likely to have more freedom to pick his destination next summer if he opts to remain at Spurs for another year.
Manchester United had expressed an interest in signing Kane this summer but have been put off by the high costs involved. They have since signed Rasmus Hojlund but would still be keen on signing Kane on a free transfer next year.
Chelsea, meanwhile, are also in the market for a new striker after pulling out of talks with Juventus to include Dusan Vlahovic as part of a swap deal with Romelu Lukaku.
Paris Saint-Germain have held talks with Kane this summer and would be interested in a move next year to replace Kylian Mbappe, who is likely to join Real Madrid on a free transfer.