Real confirmed the agreement and said the 23-year-old would sign a five-year deal with the club after undergoing a medical.
Spanish media reported that the nine-times European champions would pay around 36 million euros ($48.53 million) for the player.
Robben had been stalling over signing a contract extension at Chelsea, even though his existing deal has two years to run.
The news of Robben's signing came just a few hours after Real announced they had agreed to buy Argentine defender Gabriel Heinze from Manchester United for a reported 12 million euros.
He becomes Real's eighth signing of the close season following defenders Heinze, Christoph Metzelder, Pepe, Dutch duo Wesley Sneijder and Royston Drenthe, Argentine striker Javier Saviola and former Liverpool goalkeeper Jerzy Dudek.
With Ruud van Nistelrooy having signed for Real last season, Robben's signing brings the club's Dutch contingent to four.
Real's total expenditure since Fabio Capello was sacked following their victory in the Spanish championship is in the region of 120 million euros.
LONG-TIME TARGET
The winger has been a long-time target of Real's and together with Cesc Fabregas and Kaka was one of the players Ramon Calderon promised to sign when he became president last July.
The pacy, left-footed winger joined the Londoners from PSV Eindhoven for $22 million in 2004, and quickly established himself as a force to be reckoned with, helping them win back-to-back league titles in 2005 and 2006.
He weighed in with 15 league goals in three seasons but was dogged by inconsistency and injury and made just 51 Premiership appearances. He underwent knee surgery at the end of March and has not played since.
Robben's future was thrown into futher doubt when Chelsea moved to sign France midfielder Florent Malouda from Olympique Lyon last month, though they denied he was a replacement.
The winger will compete with Brazilian Robinho and Drenthe for a place on the left of midfield though his versatility allows him to operate on both flanks and he is perhaps more likely to act as a replacement for David Beckham down the right.
(Additional reporting by Simon Baskett in Madrid)
More from rediff