In 2023, Costa accepted a probationary sentence to avoid trial for an aggravated robbery in 2018, which US officials ruled that he is still serving.
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Criminal convictions are grounds for inadmissibility to the United States. The decision comes amid a broader crackdown by the Trump administration on issuing visas to some visitors and students.
"We can confirm Ayrton Costa is set to miss the Club World Cup with Boca," a club spokesperson told Reuters.
"The defender was due to travel on Wednesday night ... but did not obtain the visa."
The US embassy in Buenos Aires was not immediately available to respond to a request for comment emailed outside of normal business hours, it said in an automated reply.
Boca's opening game is against Portuguese side Benfica in Miami on Monday.