The German champions had their second bid for the England captain rejected following a meeting between Levy and Bayern directors Marco Neppe and Jan-Christian Dreesen earlier this week.
Bayern are not throwing in the towel, though, and Thomas Tuchel's side are now believed to have lodged their final offer, worth a club-record £86m.
Tottenham were apparently asked by Bayern to provide an answer by Saturday morning, in order for the Bundesliga giants to move onto other targets if Levy remained defiant in his stance.
However, German outlet Bild claims that Levy has ignored Bayern's ultimatum, and the Tottenham chairman has even jetted off to Miami for a two-week family holiday.
While there, the report adds that Levy may discuss the Kane saga with silent owner Joe Lewis, who recently pleaded not guilty to charges of insider trading in the United States.
Levy memorably blocked Kane from joining Manchester City in a nine-figure deal in 2021, but the 30-year-old is yet to win a major trophy in North London and has entered the final year of his contract.
Tottenham are no closer to agreeing a renewal with their all-time top scorer, who is not expected to put pen to paper on fresh terms in North London while the transfer window remains open.
However, Levy is still hopeful that Kane will commit his future to the club if Spurs demonstrate significant progress under Ange Postecoglou, but the Spurs chairman is running the risk of losing the marksman on a free transfer in 2024.
While they have been left in the dark for the time being, a separate report from The Mirror claims that Bayern are so confident of signing Kane that they have already arranged accommodation for the striker and his family.
The Bavarians have supposedly found Kane a luxury villa in an affluent area of the city, and the England international has already given the green light to a club-record £400,000-a-week contract.
Kane became Tottenham and England's all-time leading goalscorer during a stellar 2022-23 campaign at individual level, but Spurs could only muster an eighth-placed finish in the top flight.
As a result, the forward is set to be without European football for the first time in his Tottenham career next year, although he allegedly expects to remain at the club if no deal with Bayern is agreed before their Premier League opener against Brentford on August 13.
However, ahead of Kane's possible exit in 2024, Tottenham are apparently close to signing 19-year-old Rosario Central attacker Alejo Veliz as a possible long-term successor.