Oil prices fell and stocks jumped in Asian trade on Wednesday after Wall Street hit record highs on hopes that a deal could soon be reached to restore traffic through the blockaded Strait of Hormuz.
"They had an all-day negotiation," Trump told Fox News.
The Strait of Hormuz "is going to be open very soon", Trump said.
"If they back out again, they are going to get hit really hard."
Trump has often threatened massive strikes if Tehran did not reach an agreement and then cited progress in talks that have so far failed to end hostilities.
Axios, citing two regional sources and a US official, reported the US and Iran were closing in on an interim agreement to reopen the strait with the agreement of Oman, which controls part of the waterway.
The report said Washington was aiming for an announcement on Wednesday (Thursday AEST).
However, Iranian state media, citing an informed source, said an agreement with Oman on the strait, which usually carries about a fifth of global oil and liquefied natural gas shipments, would be delayed "as long as the United States continues to threaten Iran".
Trump has yet to achieve the objectives he set out at the start of the war: dismantling Iran's nuclear program, curbing its ability to attack regional rivals and creating conditions for Iranians to overthrow their clerical rulers.
Iran has halted most traffic through Hormuz while Washington maintains a blockade of Iran-related shipping and ports.
Qatar had earlier said mediators were making progress in efforts to end the conflict, even after Tehran denied that talks were under way.
Trump and Qatar's Emir Sheikh Tamim bin Hamad Al Thani discussed efforts to narrow differences between Washington and Tehran and improve the prospects for a lasting settlement during a phone call on Tuesday, Qatar's Emiri office said.
Iran's foreign ministry said talks with Oman over transit through the strait were positive and continuing, and were focused on establishing safe shipping lanes, state media reported.
The route being negotiated was aimed at ensuring the sovereign rights and national security of both Iran and Oman, it said.
Iran has long demanded control over the Strait of Hormuz, which it says it should share with Oman on the opposite shore.
But Tehran rejected an Omani proposal last week that would have allowed shared oversight and the collection of voluntary fees from ships.
Yemen's Iran-aligned Houthis have also imposed a naval blockade on Saudi Arabia in the Red Sea, further restricting oil export routes.
A projectile struck an Indian-flagged vessel near Yemen, causing it to capsize and sink, but all 14 seafarers on board were rescued, Indian officials said, in an attack Yemen blamed on the Houthis.
The United Nations' International Maritime Organisation has reported at least 17 deaths of seafarers from 64 incidents in the Strait of Hormuz since the war started on February 28.
Those are in addition to the Iranian government's toll of 50 people killed and 500 wounded by US strikes since hostilities renewed in late June, a figure reported before Iran said a US strike on Qeshm in late July killed a husband and wife and their 2-year-old son.
In all Iran has reported more than 3400 deaths since the US and Israel launched the war on February 28, while the US has reported 18 military personnel dead.
