Reports had indicated that Madrid had deposited the midfielder's contract at the offices of the Spanish Football League, having resolved to activate his €30m release clause and pay the previously disputed extra €9m due in VAT.
Real Sociedad then called a press conference to announce the move, while Real Madrid confirmed the news on their official website.
He has agreed a six-year contract and will be presented at the Bernabeu stadium on Saturday after undergoing a medical.
“I became aware of the interest whilst on holiday,” he said in the press conference, before revealing that it was a difficult decision to leave the Anoeta.
“I spoke with my people, we considered it and I thought, this is a unique and irresistible offer.
“I have been here 12 years and the decision was not easy, I had a hard time deciding. I feel the colours of La Real and will always be a fan.
“Real Sociedad has given me everything and I am what I am because of La Real. I have been here 12 years, I arrived here as a child and have matured into a man.”
The Basque club qualified for the Champions League play-offs last season. President Jokin Aperribay said: “This morning the LFP called to tell us that Real Madrid have activated Illarramendi’s buyout clause.
“We wanted him to stay [but] we want to thank the player for the years he has been here. We are sorry that he is leaving.
“We did not want to negotiate with Real Madrid, and this we also told to Asier and his agents.
“We were not prepared for the money to be paid in instalments.”
Illarramendi, 23, is seen as a long-term replacement for Xabi Alonso, another former player at Basque club La Real who will be 32 in November, and he will also compete for a place with Germany's Sami Khedira and Croatia's Luka Modric.
“I have been saturated these last few days with a lot of tension, and now I undertake a new stage that I will give everything to,” added the player.
“It will be different [at Madrid], there will be more media attention, but I will give everything in every training session.”
Big-spending Real president Florentino Perez is known for luring ‘galacticos’ like David Beckham, Zinedine Zidane and Cristiano Ronaldo to the Bernabeu - but Illarramendi's capture follows hard on the heels of the purchase of playmaker Isco from Malaga and the return of full-back Daniel Carvajal from Bundesliga side Bayer Leverkusen.
The trio were in the Spain squad that won the U21 Euro championships in Israel last month and Isco and Illarramendi were both named in UEFA's team of the tournament.
Real are looking to bounce back from a poor season under former coach Jose Mourinho, who quit and returned to Chelsea after failing to win any major silverware.
Portuguese Mourinho was replaced by Italian Carlo Ancelotti, whom Real bought out of his contract with French champions Paris St Germain.
Under construction magnate Perez, Real have spent hundreds of millions of euros on players but have failed to win the 10th European crown they have been chasing since their last continental triumph in 2002.
They are the world's richest club by income ahead of Spanish rivals Barcelona in second, according to accounting firm Deloitte.
Illarramendi's move is another reminder that it is increasingly the case that only the two heavyweights can afford the nation's best players and those who do not play for Real or Barca are seeking top wages abroad.
Read the original article on FootballEspana