Bayern chairman Karl-Heinz Rummenigge confirmed on Tuesday that negotiations had been opened up with the Champions League winners.
Kroos, whose contract with Bayern has only a year left to run on it and who has turned down their offer of a renewal, is currently holidaying in Majorca after winning the World Cup with Germany.
While on the Spanish island, he met up with his agent on Tuesday before confirming his intentions to sports daily AS.
"I'll be going to Madrid either tomorrow or the day after," Kroos said.
A fee of around 25 million euros is reportedly being demanded by Bayern for the former Bayer Leverkusen midfielder, who impressed in Brazil.
Kroos, 24, was also reported to be a target of Manchester United this summer, but Real appear to have won the race to sign him with an official announcement now expected this week.