The German champions have asked Spurs chairman Daniel Levy to make a decision by Saturday morning over Kane’s sale after they increased their bid to a ‘very fair offer’ for the striker, who has one year left on his contract.
Bayern want clarity so they can move on to other targets if their bid is unsuccessful.
It is now down to Levy to decide whether to accept Bayern’s club-record offer of £86.24m plus add-ons, with all parties wanting the situation resolved quickly.
Bayern hope to have Kane in their line-up for next Saturday’s German Super Cup final against RB Leipzig and know the player is eager to accept their challenge and sign a four-year contract worth around £400,000 a week.
If a deal cannot be agreed, Kane is expected to see out the final year of his contract at Tottenham and leave for free next summer.