The Bundesliga champions are aiming to seal the club record signing of Kane within the next 48 hours having agreed a fee of €100m (around £84.5m) plus add-ons with Tottenham.
Spurs refused to rule out Kane staying despite accepting the German giants' latest offer, but have left the final decision to the England captain.
Bayern, who have offered Kane a five-year contract, are said to have been concerned that the England captain was having second thoughts but are now optimistic he will sign on the dotted line.
It is even thought he will wear no.9 in Germany, the number worn by Robert Lewandowski for so many years and who scored 344 times for the club.
Kane, 30, has entered the final year of his Tottenham deal and recently rejected new terms worth £400,000-a-week.
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Spurs are reluctant to lose the England captain to a Premier League rival for free next summer and so have agreed a deal with Bayern who it is said has agreed to pay the bulk of the money up front.
Bayern made their first bid for Kane in June and Spurs chairman Daniel Levy rejected three offers from the German champions before accepting the fourth.
They even went public in their pursuit of Kane, with Herbert Hainer, the chairman, acknowledging negotiations between all parties. "Bayern always try to bring in stars," Hainer told Kicker, the German football magazine.
"Harry Kane is a highly attractive player, the England captain and a top scorer. He would do us and the Bundesliga good."
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Manchester City were ready to pay £127m to sign Kane in 2021 but Levy stood firm and rejected their advances.
Despite Kane's continued heroics since then, the 280-goal striker has been left without a major trophy to show for his 19 years at Tottenham.
Spurs do not play in any European competitions this year, while Bayern will be among the favourites to win the Champions League.
Kane has been associated with Spurs since he was 11-years-old, making his first-team debut in 2011 but has just League Cup and Champions League runners-up medals to show for it.
After a series of loan spells he broke into the first team in the 2014/15 season and has gone on to become one of their greatest ever players.
As well as being Tottenham's record goalscorer, Kane is out there on his own for goals with England having so far netted 58 times for his country.
His last appearance looks to have been scoring four goals in a 5-1 friendly win over Shakhtar Donetsk last Sunday.
Currently, he has scored 213 Premier League goals and is 48 behind Alan Shearer's record of 260, but will be 35 if he sees out his contract in Munich.