The German champions said they had reached agreement with Deportivo Coruna and last season's Spanish first division top scorer with 29 goals would sign a four-year contract for an undisclosed transfer fee.
The Bavarians said in a statement the 28-year-old would have a medical on Tuesday and could make his debut in a Bundesliga game at Hanover 96 on Saturday.
Bayern had been chasing the Dutchman for weeks but were fighting with his Spanish club over the transfer fee.
Bayern's commercial manager Uli Hoeness said last weekend that Makaay had agreed to join the four-times European champions but Deportivo were asking for 20 million euros ($22.61 million) while Bayern were prepared to pay 16 million.
The Munich club, who completed their fourth domestic league and Cup double last season, were looking for a top striker in a bid to make up for their shock first-stage exit from the Champions League last year.
Bayern launched the defence of their German title with a 3-1 home victory over promoted Eintracht Frankfurt on Friday.
Former Bayern and Germany great Franz Beckenbauer, now the club's supervisory board chairman, had earlier weighed in by saying the Germans should accept Deportivo's conditions.
TIME RUNNING OUT
"I am certain that we need a third top striker alongside Elber and (Claudio) Pizarro for a long season," said Beckenbauer.
"If we believe that, then we have to win this poker game," the World Cup-winning captain and coach wrote in a column in Monday's top-selling Bild daily.
"Whether we have to pay a million more is not important. Such a player would be an enrichment for Bayern and the entire Bundesliga."
Hundreds of Bayern fans had also sent e-mails to the club to say Makaay was definitely worth having even if it meant a financial effort.
The deal means Bayern will be able to field him from the start of their Champions League campaign.
Time was running out as Deportivo are set to play in the third qualifying round of the Champions League on August 13.
"If Deportivo register him for the Champions League, we can forget about the transfer," Bayern chairman Karl-Heinz Rummenigge had said this weekend.
Brazilian Elber and Peru striker Pizarro have been Bayern's first-choice attack. Makaay's arrival means they now have to fight to keep their places in the starting line-up.
Bayern know Makaay well since he scored a hat-trick in their 3-2 defeat to Deportivo at the Olympic stadium in their opening game of last season's disastrous Champions League campaign.
