Marca published a disparaging article about the 22-year-old following his £30 permanent switch to Arsenal, describing him as "an economic bonanza but a sporting failure".
But Odegaard fired back at the Spanish media on Friday, hours after his move back to the Gunners had been officially confirmed.
He posted to social media: "More than six and a half years ago I signed for Real Madrid and fulfilled a huge dream in being able to wear the white shirt, play at the Santiago Bernabeu and play in the Champions League.
"I couldn't be more proud. I came here aged 16 and I had the opportunity to learn from the best in the world and my idols. I've learned a lot and enjoyed myself along the way. I'm grateful to everything that happened to me and everything I've learned.
"I want to thank all the people who've helped me improve and grow at Real Madrid, be they coaches, teammates, physios, club employees and fans. Thank you!
"I've been through some good moments and some bad ones, as happens often in life and in football. I've become much stronger and more prepared for the future.
"These days a lot has been said about me and my motivations for leaving. I only want to say that almost everything that is being said is not true. It's a real shame that the press say things that aren't true, because many people sincerely think that's how it is.
"I have my reasons, I know what the truth is and what happened and that's why I think this leaving is for the best."
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