Tehran has already announced it will stop U.S. citizens entering the country following Washington's decision to ban refugees and citizens of seven majority-Muslim countries including Iran.
"A special committee in Tehran reviewed their cases and decided to oppose the visit by the U.S. freestyle wrestling team," Foreign Ministry spokesman Bahram Ghasemi was quoted by state television as saying.
"The visa policy of the new American administration gave us no other option but to ban the wrestlers."
USA Wrestling said that neither they nor United World Wrestling have been officially notified that these reports are accurate.
"We have no official documentation that the USA Wrestling team will be denied the opportunity to participate at the Freestyle World Cup," USA Wrestling said in a statement.
"The Freestyle World Cup is the annual World dual meet championship, one of the most important events that United World Wrestling hosts each year.
"If these reports are true, USA Wrestling is extremely disappointed about this, which we believe would be an unacceptable situation. Wrestling is about competition and goodwill through sport, and is no place for politics."
USA Wrestling said on Monday the United States would send a team to Iran, where U.S. freestyle wrestlers have participated in competitions since 1998.
Nenad Lalovic, president of United World Wrestling, the sport's international federation, said it was working towards a solution.
"Though we had hoped for a different outcome from Iran's Foreign Ministry, we appreciate the complex nature of this decision. We are currently working to find a solution for the Freestyle World Cup as soon as possible," he said in a statement.
"Despite the travel restrictions between their countries I've been encouraged this week by the cooperation and friendship of the Iranian Wrestling Federation and USA Wrestling.
"As we've seen over the years, wrestling is a sport that unites people and nations. United World Wrestling hosts a full calendar of events every season and we're certain to see American and Iranian wrestlers on the same mat again in the very near future."
Trump has adopted an aggressive stance towards Tehran since Iran's ballistic missile test-launch, saying it was now "on notice". The new U.S. administration is expected to impose fresh sanctions on multiple Iranian entities.
(Writing by Parisa Hafezi; Editing by Tom Heneghan and Toby Davis)
