The US president also reportedly told aides he expects to resume bombing after the November congressional elections,
Iran's Foreign Minister Abbas Araqchi had told reporters at the United Nations on Friday that Iran conveyed its peace plan to the United States through Qatari mediators, saying a deal would trigger a seven-day countdown to reopening the strait and a pause in regional fighting.
"If there is seriousness on the US side to come to a deal and reopen the Strait of Hormuz, everything is now prepared," Araqchi said.
A US official had told Reuters the US and and Iran were having "positive and constructive discussions through the mediators," but Trump was privately sceptical that Iran would meet his demands, the Journal report on Friday said.
He has told his staff he sees a renewed bombing campaign as likely.
Reuters could not immediately confirm the report.
Diplomatic efforts surrounding the UN General Assembly in New York underscored the global stakes of the seven-month-old conflict, which has disrupted oil flows through one of the world's most critical shipping lanes and drawn in regional and world powers.
If the US were to agree, the warring parties could start wider talks "on mutually agreed subjects," Araqchi said.
Tehran would reopen the Strait of Hormuz and hostilities would end on all fronts, including Lebanon. The plan in turn requires the US to lift its blockade on Iranian ports, release Iran's frozen funds, and waive its oil sanctions.
The Iranian seven-day proposal comes after previous attempts at diplomacy have failed to end the war, which began on February 28 with US and Israeli strikes.
Since then the war has alternated between pauses and renewed strikes, killing thousands while degrading Iran's conventional forces and damaging Iran's already ailing economy.
Iran and its allies have retaliated by extending the conflict throughout the region. Iran has struck US allies in the Gulf with drones and missiles while its Houthi allies in Yemen have targeted Saudi interests.
Iranian-aligned militias in Iraq were blamed for a pipeline attack that temporarily took more Saudi oil off the market. All the while, Iranian-backed Hezbollah has been engaged with Israeli forces in Lebanon.
With the Saudi-backed, internationally recognised Yemeni government under assault from Iran-linked Houthi fighters, and Houthi missiles striking inside Saudi Arabia, allies of the kingdom have united to show support.
Defence officials of Turkey and Pakistan, both partners in a joint defence pact with Saudi Arabia, met in Riyadh on Friday to review the security situation and discuss intelligence cooperation, military coordination, and integrating forces, the Saudi defence ministry said in a statement.
The statement came a day after Saudi, Pakistani and Turkish foreign ministers met within the framework of the Mecca Joint Defence Agreement.
The violence in Yemen has prompted a new humanitarian crisis, the United Nations children's agency said on Friday, reporting a rise in acute malnutrition among children.
