"US diplomats don't roam around Tehran, so we don't see any reason for Iranian diplomats to roam freely around New York City, either," Pompeo told the newspaper.
"It's absolutely appropriate that we provide Foreign Minister Zarif and his delegation with all the rights that they are due under the UN headquarters agreement, and nothing more than that," he added.
The Iranian Foreign Minister is in New York to attend a meeting of the UN Economic and Social Council, a high-level ministerial meeting on sustainable development goals for issues that include conflict, hunger, gender equality and climate change, according to The Washington Post.
His visa was issued at the US Embassy in Bern, Switzerland on Sunday, a day before he arrived in New York.
The US has an agreement that requires it to issue visas expeditiously to all foreign diplomats travelling to New York for UN events.
Ties between Washington and Tehran have deteriorated in recent months amid a trade of barbs, military disputes and continuous US sanctions against Iran.
Earlier on Sunday, Iranian President Hassan Rouhani said Iran was ready to hold negotiations with the US if Washington lifts sanctions against Tehran.