Spurs have offered a new contract but Kane, 29, is not willing to extend his deal with the Premier League club, sources added.
Bayern have already made two bids to Spurs for the England captain -- the second one worth €94 million ($103.19m). The club's hierarchy could be tempted to submit an improved bid to get the deal over the line.
Kane becomes a free agent when his Spurs contract expires at the end of next season but he wants to sign for Bayern this summer.
Sources have told ESPN that Bayern coach Thomas Tuchel met Kane in London a few weeks ago and the striker was attracted to the idea of winning trophies with the German giants.
According to sources, Kane's brother and agent, Charlie, has been in constant contact with the Bayern hierarchy ahead of a potential transfer.
New Spurs manager Ange Postecoglou said Wednesday that he could not give any assurances around Kane's future but added he wanted to talk to the player once he returned from his holiday.
If Kane outlines his desire to leave the club, Spurs chairman Daniel Levy will have to decide if they want to risk losing the player for free next year or if they are ready to lose him now for a substantial fee.
Reports in England have claimed that Spurs could let him go for £100m this summer but ESPN sources added Bayern would be reluctant to pay that figure.
According to sources, Paris Saint-Germain are also monitoring the situation with the club interested in signing Kane. The French champions are waiting to see how negotiations play out between Bayern and Spurs before making a move for the striker.