Zorreguieta is believed to have taken her own life.
The King of the Netherlands, Willem-Alexander, and Queen Maxima are flying to Argentina for the funeral.
The royal house said that the Queen is "very shocked and very sad" by the news.
The Dutch prime minister, Mark Rutte, said: "The sudden death of Queen Maxima's sister comes as a big shock".
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She was a civil servant who had been working for the Argentinian government since 2016, in an office that oversees social policies.
According to reports, Inés Zorreguieta had been suffering from depression.
Tributes were paid to the "very handsome and talented" Zorreguieta on a video her brother Martin posted of her singing and playing the guitar a year ago.
Zorreguieta died late on Wednesday, aged 33, a palace spokeswoman announced earlier.
She was a bridesmaid for her sister's 2002 marriage to then-Dutch Crown Prince Willem-Alexander, and goddaughter for the couple's youngest daughter, 11-year-old Princess Ariane. I wish them all the strength necessary to carry this frightful loss together. Dutch News reported she canceled a visit to Lithuania, Estonia and Latvia following the news of her sister's passing.