BBC - Lionel Messi's father is currently dealing with a "health-related situation", the family said in a statement on Thursday.
"At this time, he is under medical supervision, recovering and progressing favourably within the condition he is experiencing," the statement said.
Messi was in tears after scoring his first goal in Argentina's 3-0 victory over Algeria in their opening World Cup match earlier this week.
Argentina's captain, 38, went on to complete a hat-trick, drawing level with Miroslav Klose at the top of the all-time World Cup goalscoring list.
"Why did I cry? It was something completely unrelated to football," Messi said after the match.
"I went through some difficult days, but I'm grateful to the entire delegation and my team-mates because they were always by my side, giving me a lot of strength to help me get through it."
The family addressed the speculation around Jorge Messi's health following that opening game.
"In light of speculation that has circulated in recent hours, the family wishes to express its deep discomfort with the lack of sensitivity, respect, and discretion with which some people have treated what is strictly a private family matter.
"The family also wishes to clarify that only his closest relatives have real and accurate information regarding Jorge's condition. Therefore, any version, statement, or information that does not come directly from the family and its official channels should not be considered valid or truthful."
The statement continued: "At times like these, we ask for responsibility, prudence, and humanity. A person's health and the peace of mind of those around them should not be the subject of speculation or irresponsible media interest.
"We sincerely appreciate the expressions of affection, respect, and concern we have received, and we ask that Jorge's privacy, confidentiality, and intimacy - as well as that of his entire family - be respected throughout this process."
Argentina's next group game is against Austria on Monday, June 22, before their final group game against Jordan on Sunday, June 28.