Paolo Zampolli, an Italian-American who is an envoy for global relations, told the Financial Times he made the suggestion to Trump and FIFA President Gianni Infantino.
"I'm an Italian native and it would be a dream to see the Azzurri at a US-hosted tournament. With four titles, they have the pedigree to justify inclusion," said Zampolli, who has no official connection with the World Cup or Italian football.
The plan seems to be an effort to repair ties after Trump and Italy's Prime Minister Giorgia Meloni fell out amid the US leader's attacks against Pope Leo XIV over the Iran war.
Italy's main sports news websites gave the story only a passing reference but politicians and officials were quick to reject the idea.
"First of all, I don't think it's possible," Italian Olympic Committee President Luciano Buonfiglio said. "Second, I'd feel offended. In order to go to the World Cup, you have to earn it".
Sports Minister Andrea Abodi echoed that, saying "it is not appropriate... you qualify on the pitch," while Economy Minister Giancarlo Giorgetti described the concept as "shameful".
Soccer's world governing body FIFA responded by pointing to Infantino's previous comments on Iran's participation.
"The Iranian team is coming, for sure," he told last week's CNBC Invest in America Forum: "They really want to play, and they should play. Sport should be outside politics."
Currently there is no suggestion that Iran will withdraw or be banned from the tournament, which Italy missed out on after losing in a playoff for the third World Cup in a row.
Iran qualified for a fourth successive World Cup last year but, after the start of the war, requested that FIFA move the team's three group matches from the US to Mexico - which was rejected.
Iran are seemingly proceeding as planned. "We are preparing and making arrangements for the World Cup, but we are obedient to the decisions of the authorities," Iranian football federation President Mehdi Taj told reporters at a pro-government rally in Tehran on Wednesday.
