Both countries' presidents were due to attend the annual United Nations General Assembly in New York. Trump is scheduled to address the gathering at 10:15am (1415 GMT).
Foes for decades, Washington and Tehran have been increasingly at odds since May, when the Republican US president pulled out of the 2015 international nuclear deal with Iran and announced sanctions against the OPEC member.
The accord, negotiated under Democratic US President Barack Obama, lifted most international sanctions against Tehran in exchange for Iran curbing its nuclear program.
Over the summer, Trump had said he would meet with Rouhani without preconditions to negotiate a new deal, an offer reiterated on Sunday by US Secretary of State Mike Pompeo and extended to Iran's supreme leader, Ayatollah Khamenei.
Rouhani said on Monday Tehran would not talk to Trump until the United States returned to the 2015 deal..
The top adviser to Khamenei, Ali Akbar Velayati, rejected the US offer on Tuesday, saying "Trump's and Pompeo's dream would never come to reality," the IRNA news agency said.
"Despite requests, I have no plans to meet Iranian President Hassan Rouhani. Maybe someday in the future. I am sure he is an absolutely lovely man," Trump wrote in a post on Twitter.
He said the Iranians have no choice but to change before a meeting with him can occur, but he looks forward to having a great relationship in the future.
"I'm not meeting with them until they change their tune. It will happen. I believe they have no choice. We look forward to having a great relationship with Iran, but it won't happen now," he said shortly before delivering his speech.
Alireza Miryousefi, spokesman for Iran's UN mission, told Reuters that Iran has not requested a meeting with Trump.
Some Iranian insiders have said any talks between Rouhani and Trump would effectively kill the existing nuclear accord.
Tensions have been rising after a deadly attack on a military parade in southerwestern Iran in which 25 people were killed. Khamenei said on Monday the attackers had been paid by US ally Saudi Arabia and the United Arab Emirates, and that Iran would "severely punish" those behind the bloodshed.
Quashing the current pact would come at a political cost for the Iranian president, who championed the deal with the supreme leader's guarded backing and could lose support from European allies.
Rouhani is also under increasing pressure from Iranian hardliners, including Iran's elite Revolutionary Guards, which have kept up the anti-American rhetoric ahead of the UN session.
Trump's administration is pushing allies to cut imports of Iranian oil to zero as Washington prepares to restore sanctions on Iran's oil sales in November.
The remaining countries in the deal, which see it as the best chance to stop Iran from developing a nuclear bomb, on Monday agreed to keep working to maintain trade with Tehran.
German Foreign Minister Heiko Maas, following a meeting on Monday with Britain, China, France, Germany, Russia and Iran in New York, warned that the US strategy of applying maximum pressure on Tehran and going it alone could risk a regional escalation.