The winger will not feature for his country at the U20 World Championship, even though boss Javier Mascherano was reportedly calling for him to do so.
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Garnacho has recently been struggling with injury and hasn't featured for United since the middle of March as a result.
However, The Mirror report that he was desperate to play in the tournament, which commences in Argentina from May 20.
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Ten Hag wants to keep the 18-year-old at United for their final games of the season, but Mascherano pleaded with him to let him go.
The former Liverpool and West Ham star likened the situation to that of Lionel Messi in 2008 for the Beijing Olympics.
Messi was at first not allowed to join the Argentina squad for the tournament, but Barcelona eventually allowed to travel to China.
He scored two goals on their way to winning the gold medal, before lifting La Liga and the Champions League with his club in the subsequent months.
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Mascherano is said to have told Ten Hag that such an experience helped turn Messi into the superstar that he is now.
However, it wasn't enough to change the mind of the Dutchman, who is keen to keep Garnacho for the final weeks.
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The youngster has had an impressive season thus far, scoring four goals and assisting a further five in all competitions.
Garnacho's efforts have been rewarded with a new contract at United until 2028.
He is yet to make an appearance for the Argentina national team, but this is very likely to change in the coming seasons.