Alaba, who played in all of Austria's Euro 2020 matches, agreed to join Madrid following the expiration of his contract with Bayern Munich, where he had spent his entire career, winning 27 trophies.
The 29-year-old – who has taken Madrid's number four shirt vacated by Sergio Ramos – was officially unveiled at the Santiago Bernabeu on 22 July (AEST).
However, his pre-season preparations have been hit by a positive COVID-19 test.
Madrid revealed the news in a brief statement on their official website. They did not clarify whether Alaba was asymptomatic.
Alaba, who has signed a five-year deal with Los Blancos, is the second Madrid player to test positive in as many weeks, after Karim Benzema also contracted the virus.
With Madrid having agreed to sell Raphael Varane to Manchester United, Alaba is set to lead a new-look defence this season.
Carlo Ancelotti's team were beaten 2-1 by Rangers in a pre-season friendly on Sunday – Alaba did not feature, but has been training with the squad.