ESPN claims that the initial payment would be €3 million, with a possible variable payment of just over €1 million.
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The 28-year-old played just four times this season, and club management have accused him of postponing his injury recovery so he could play in the World Cup.
Depay was permitted to leave his previous team on Wednesday to finalize his contract, and he participated in his first Atletico training session on Thursday.
After falling down the pecking order under Xavi’s leadership, the striker’s contract at the Camp Nou was due to expire at the end of the current campaign with no option for renewal.
Following the club’s approval for fugitive talent Joao Felix to join Chelsea on a season-long loan for £9 million, Depay has agreed to join Real Madrid.
The Dutch striker has changed clubs five times since making his professional debut at PSV Eindhoven as a young player, most recently to the capital of Spain.