Real did not give details of financial terms but Spanish media reported they had agreed to pay around 3 million euros ($3.5 million) for the 16-year-old.
He will be presented to the media at Real's training base outside the Spanish capital on Thursday, the world and European champions said on their website (www.realmadrid.com).
Odegaard made his debut for Norwegian top-flight side Stromsgodset at the age of 15 and the playmaker's precocious talent has also attracted the attention of some of Real's fellow European heavyweights like Barcelona and Bayern Munich.
He became Norway's youngest international when he started against United Arab Emirates at the age of 15 years and 253 days in August, breaking a record that had stood for more than a century.
(Additional reporting by Philip O'Connor in Stockholm, writing by Iain Rogers, editing by Patrick Johnston)
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