No fee was revealed for the transfer of the Courtois, who had just one more year to run on his Chelsea contract, although Spanish media claimed that the deal is worth around 35 million euros.
The English giants have been forced into finding a replacement for Courtois, 26, after he failed to report for pre-season training on Monday.
And with his two children still resident in the Spanish capital, Courtois's agent claimed earlier this week family commitments were a motivating factor in forcing a move to the European champions.
"He has made it clear to the club that the best option for him is to move to Madrid," Christophe Henrotay told the Sun newspaper.
"For him this is a big decision because he wants to be near his family, and there is an offer in place for Chelsea to accept."
Just weeks after Liverpool broke the record transfer fee for a goalkeeper in a deal worth up to 72.5 million euros for Brazilian international Alisson Becker, the inexperienced Arrizabalaga, 23, is now set to inherit the title of the world's most expensive goalkeeper.
Earlier on Wednesday, Bilbao confirmed that Arrizabalaga had paid his buyout clause at the league's headquarters in Madrid to free him from his contract with Athletic. Arrizabalaga has played 53 times in La Liga for Bilbao over the last two seasons, and made his international debut in a 5-0 friendly win over Costa Rica in November 2017.
But for a late change of course due to the wishes of former Real boss Zinedine Zidane, Arrizabalaga could have joined Madrid himself in January when his buyout clause was just 20 million euros.
However, after that move collapsed, he signed a new seven-year deal with the Basque side that dramatically increased his release clause and has cost Chelsea dear.
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)