Balogun was set to miss Monday's last-16 knockout clash with Belgium having picked up a red card in his previous match, but FIFA on Sunday suspended the ban for a year, making the forward eligible to play.
Sources told AFP Trump personally called FIFA President Gianni Infantino, asking him to review Balogun's punishment.
Former Liverpool manager Klopp, who is set to become the next Germany manager after Julian Nagelsmann's resignation on Friday, hit out at the decision.
"It's a red card, even if it's a shame because Balogun can't play, but it's there in the rules," Klopp said.
"This is our game, not theirs... If Trump and Infantino really worked this all out between themselves, that's crazy. It calls everything into question...
"These two individuals, neither of whom has a clue about football, shouldn't have anything to do with it," the 59-year-old added.
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Balogun, 25, was shown straight a red card after a VAR review showed him stepping on the foot of a Bosnia defender in the US's 2-0 last-32 win.
FIFA's regulations state a red card automatically triggers a one-match ban, which cannot be appealed.
But a rule in the FIFA statues allows the governing body to "fully or partially suspend the implementation of a disciplinary measure", essentially amounting to a veto over refereeing decisions.
The previously little-known rule was the basis for overturning a three-game suspension imposed on Cristiano Ronaldo in the lead-up to the tournament which meant the superstar forward was eligible to play in Portugal's first two games.
German FA (DFB) president Bernd Neuendorf demanded an explanation from FIFA, saying the decision jeopardised the tournament's credibility.
"FIFA should now quickly address reports suggesting that the decision to suspend the red card issued to American player Folarin Balogun was preceded by a phone call between US President Donald Trump and FIFA President Gianni Infantino," Neuendorf told AFP subsidiary SID on Monday.
"The impression that there was active political interference in the sport must be swiftly and conclusively dispelled. It is a matter of the integrity of the competition and the credibility of FIFA."
Trump, who was the inaugural recipient of the newly created FIFA Peace Prize in the lead-up to the tournament, praised FIFA's decision on social media.
"Thank you to FIFA for doing what was right and reversing a great injustice!" Trump wrote on his Truth Social platform.
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