The 28-year-old netted 42 goals for the Bundesliga champions last season in all competitions.
The 28-year-old netted 42 goals for the Bundesliga champions last season in all competitions and German daily Bild estimates he has been rewarded with a new deal worth over 15 million euros ($15.9m) per year.
"I am very pleased that the negotiations have come to a positive conclusion and I will be bound to FC Bayern for a few years," said Lewandowski.
The target-man has scored 19 goals so far this season including two in the 5-0 win over Wolfsburg that sent Bayern back to the top of the Bundeslga at the weekend.
"We have the same big goals."
Lewandowski joined Bayern in 2014 on a free contract from Dortmund.
Since then he has scored 89 goals for Bayern in 122 matches, which includes scoring 58 times in 77 Bundesliga games.
He was the Bundesliga's top scorer last season with 30 goals, having also won the award in 2013/14 with Dortmund when he scored 20 goals.
Lewandowski has a market value of around 50 million euros.
Having reached the semi-finals of the Champions League for the last two seasons running with Bayern, Lewandowski has made no secret of his desire to win the European title with the Bavarians.
He finished on the losing side for ex-club Borussia Dortmund when the Bavarian giants won the 2013 final in London.
"There are new challenges facing us! Mia San Mia!," Lewandowski wrote on Twitter, quoting the Bayern's motto 'we are who we are'.
The signature of the Poland hot-shot, who had been linked to Spanish giants Real Madrid in the past, is a significant milestone for Bayern, who now have ten players all with a contract until 2020 or beyond.
"I'm happy we've secured the services of one of the world's best centre-forwards over the long term," said Bayern's chairman Karl-Heinz Rummenigge.
Behind the scenes, Bayern had been working for months to secure Lewandowski's services.
"He's one of the best strikers in the world, it's as simple as that," said Bayern coach Carlo Ancelotti.
With Lewandowski's signature in the bag, Arjen Robben and Xabi Alonso are the only top names with a Bayern contract due to expire at the end of the season.
Earlier on Tuesday, Lewandowski mocked the Ballon d'Or on Twitter after he finished only 16th in the voting as Cristiano Ronaldo won the award for the fourth time.
He wrote "cabaret" then '@francefootball', who organise the award, followed by pictures of smiley faces crying with laughter on his Twitter account.