The offer is reported to be in excess of €100 million (£86.4 million) and the decision now rests on the England captain as to whether he pushes through with the move.
The 30-year-old forward - who has a year left on his contract with Spurs - was previously said to want his future decided prior to Tottenham's Premier League opener against Brentford on Sunday.
He was notably absent from the Tottenham squad during their final pre-season friendly against Barcelona on Tuesday.
Bayern Munich have held a long-standing interest in Kane but have seen multiple bids knocked back by Spurs chairman Daniel Levy throughout the summer.
Kane was initially thought to be keen on a move in order to add silverware to his name but in recent weeks has also been said to be enjoying life working with Tottenham's new head coach Ange Postecoglou.
England and Tottenham's all-time leading goalscorer's partner is also expecting another child and the striker is said to be having second thoughts about leaving the country at the present time.
Sky Sports News have suggested the player is leaning towards staying on for another season but is yet to reveal a final decision.
"Bayern Munich and Tottenham have agreed a deal for England captain Harry Kane.
"Kane has a decision to make. He is leaning towards staying with Tottenham but it's ongoing," Sky Sports stated.
The striker has reportedly already agreed personal terms earlier this summer with Bayern.
Bayern had been on the lookout for a striker since the departure of star forward Robert Lewandowski to Barcelona at the end of the 2021/22 season.
Bayern won the Bundesliga in each of the Pole's eight seasons in Munich, with Lewandowski scoring 344 goals in 375 appearances.
The Bavarians struggled in Lewandowski's absence last season.
During his eight years in Munich, Lewandowski averaged 30 goals per season. In 2022/23, not one Bayern player hit the 15 league goal mark.
They were eliminated by eventual winners Manchester City in the quarter-finals of the Champions League and won the Bundesliga title on goal difference after a final-day capitulation from Borussia Dortmund.
Kane has scored more than 20 league goals in all but three of the past nine Premier League seasons, winning the golden boot on three occasions.
Kane is Tottenham's all-time record goalscorer with 280 goals in 435 appearances, having scored his first Spurs goal in December 2011.
He has scored 213 goals in 320 Premier League games and is currently 48 goals off breaking Alan Shearer's record as the highest scorer in Premier League history.
