Bayern Munich striker Harry Kane arrived in Germany on a record-breaking transfer and his sensational goal-scoring run this season has already exceeded expectations and turned him into an instant fan favourite.

Kane joined Bayern from Tottenham Hotspur in August for a Bundesliga record € 100 million ($ 106.81 million) fee, with the German champions desperate to bring in a tested scorer to fill the goal-scoring gap created by last year's departure of Robert Lewandowski to Barcelona.

Kane, the England team's top scorer and the Premier League's second all-time scorer, hit the ground running and has already bagged 21 goals in 15 matches in the Bundesliga and Champions League for Bayern.

"It is impressive," beamed Bayern sports director Christoph Freund after Saturday's double in their 4-2 victory over Heidenheim.

Kane's two goals on Saturday took his league tally to 17 from 11 games, setting a league record for most goals at this stage, beating Lewandowski's previous mark of 16.

"I cannot tell you anything different," Bayern coach Thomas Tuchel said of Kane's scoring run on Saturday. "The performances are telling a story. There is no need for any words."

Kane has also scored four times in four Champions League group matches.

"I believe he can do anything," said Bayern teammate Thomas Mueller. "We only need to bring him into position."