In Football Transfer News, the German champions have reportedly made at least one offer for Kane, who Bild says has already agreed terms.
The deadline is a signal that Bayern are growing frustrated with the current situation, although it is not clear if they would then walk away from any deal if it was not met.
Tottenham have refused to comment.
Kane, 30, is Spurs' all-time top scorer but has one year left on his contract.
There has been no suggestion Kane would be willing to extend his contract, so if Tottenham chairman Daniel Levy opts to keep the England striker it would effectively be ending the club's last chance to cash in on their talisman.
Bayern officials met Levy earlier this week but no agreement was reached.
In other football transfer news, the German champions have been told by Kane that he is ready to join but a transfer rests on the response of the Spurs chairman, Daniel Levy. He dismissed out of hand Bayern's opening offer of €70m (£60m) plus add-ons for Kane and made clear their next proposal of €80m plus add-ons was also insufficient. Bayern delegates were in London for talks this week and have made a new proposal.
Bayern's honorary president, Uli Hoeness, expressed confidence last month that the club would land Kane, saying that unless the England striker changed his mind about wanting the move "Tottenham will have to buckle".
In football transfer news, Kane has one year on his contract and has been offered a massive new deal by Spurs, significantly improving his basic £200,000-a-week terms. He regards Bayern as an attractive destination given the club's history of securing trophies – they have won 11 Bundesliga titles in a row – and being competitive in the Champions League. The club's previous record outlay was the €80m they paid to bring Lucas Hernández from Atlético Madrid in 2019.
Tottenham open their Premier League campaign at Brentford on 13 August. Bayern begin their season with a German Super Cup meeting with RB Leipzig a day earlier.
The transfer window does not close until 1 September, but it is felt all sides would prefer a resolution before the new season begins.