The two countries, which ended a first round of negotiations in Switzerland on Monday, also offered conflicting accounts about financial incentives for Iran, control of the Strait of Hormuz, and Israel's parallel war in Lebanon - all major aspects of their framework deal signed on June 17 aiming to end the war.
Nevertheless, Trump said negotiations with Iran were going smoothly: "We're getting along quite well," he said at a rally in Pennsylvania on Tuesday.
In signs of withering domestic support for the war, Trump's poll numbers weakened while the Republican-controlled Senate defied the president and voted to halt the war, in a largely symbolic move that highlighted fissures in his party.
A Reuters/Ipsos poll found 35 per cent of Americans think the US is in a weaker position with Iran than before the war, while 23 per cent believe it is in a stronger position.
The Senate vote of 50-48 endorsed a resolution passed by the House of Representatives in June, reflecting growing concern even among some of Trump's Republicans about the unpopular conflict that began on February 28.
It was the first time both chambers of Congress had passed a resolution directing a president to remove US armed forces from hostilities under the War Powers Act, though it was not clear how the votes might affect the conflict.
Though peace prospects are far from certain, the initial agreement between Washington and Tehran has allowed traffic to flow again through the strait, which typically handles one-fifth of global energy supply.
The agreement calls for Iran to allow traffic to flow freely for 60 days, though it has said it might impose tolls or other fees on shipping subsequently.
In a joint statement, Iran and Oman, which controls the other side of the strait, stressed their "sovereign rights" in the waterway, adding they would work together to manage traffic, along with associated costs.
US Secretary of State Marco Rubio, visiting Gulf allies unsettled by the peace deal, said Iran would not be allowed to charge tolls in the strait as part of any final agreement.
The deal calls for an immediate end to the war, including in Lebanon, lifting US sanctions on Tehran and unfreezing Iranian assets held abroad.
It also outlines a $US300 billion ($A434 billion) investment fund for the Iran's reconstruction.
The framework itself sets no limits on Iran's nuclear program, an issue to be tackled in 60 days of negotiations.
Trump claimed Iran had agreed to allow international inspectors indefinite access to its damaged nuclear sites.
"Iran has fully and completely agreed to highest level Nuclear inspections long into the future (Infinity!!!)," Trump said on social media.
Iran denied it had discussed its nuclear program and said it had not agreed to invite International Atomic Energy Agency inspectors back to the country.
The two sides also disagreed on details of a provision that would give Iran access to funds that have been frozen in overseas accounts.
Trump said any unfrozen assets would be used to buy food and medical supplies from the US, while Iran's ambassador to the United Nations in Geneva, Ali Bahreini, said Iran would decide how to spend that money.
Washington has already agreed to waive sanctions on Iran for 60 days, allowing Tehran to sell oil and related products and receive payment for them.
Israel's parallel war against Iran-backed Hezbollah in Lebanon also remains a sticking point.
Bahreini said the deal required Israel to withdraw its troops from Lebanon, while Israel has said it will maintain a security zone in southern Lebanon and act to "neutralise" threats against Israeli soldiers and citizens.
