But the agreement is likely to be contingent on the US lifting its blockade on Iran's ports.
Iran has insisted on some measure of control over the strait, saying it will not go back to being an open international waterway, as it was before the war.
Iran's Foreign Ministry spokesman, Esmail Baghaei, said an agreement with Oman is in the "final stage" of drafting, and a joint statement will be issued "if certain parties do not obstruct this process," apparently referring to the US.
Iranian Deputy Foreign Minister Kazem Gharibabadi also said an agreement on new shipping arrangements was nearing completion, according to IRNA.
Iranian and Omani negotiators have finalised the draft deal to reopen the strait and were awaiting approval from Iran's Supreme Leader Ayatollah Mojtaba Khamenei, two regional officials told The Associated Press.
Khamenei is believed to have been wounded in the war's opening strikes and has not been seen in public since then.
The officials, who spoke on condition of anonymity, said the potential deal will pave the way for the US and Iran to resume negotiations on Tehran's nuclear program.
Earlier, Trump said there had been an "all-day negotiation" with Iran, characterising the talks positively.
"They had an all-day negotiation," Trump told Fox News.
"We are having very good discussions," he added.
The Strait of Hormuz "is going to be open very soon," Trump said, warning that if that did not happen, Iran would be hit hard.
"If they back out again, they are going to get hit really hard," he said.
Trump was asked by reporters about a report on the Axios news site that an announcement on the Strait of Hormuz could be made on Wednesday.
"It could happen. Tomorrow or the next day," Trump said. "A lot of progress has been made."
Iran blocked the Strait of Hormuz, a main route for energy supplies from the Middle East, after the start of the war.
The Iran war began on February 28 when the US and Israel attacked Iran and Iran responded with its own strikes on Israel and Gulf countries that host US bases.
US-Israeli strikes on Iran and Israeli attacks on Lebanon have since killed thousands and displaced millions.
During an address to supporters in Las Vegas, the president expressed a preference for diplomacy to prevail.
"I'd rather make a deal because I don't want to kill people. I do not want to kill people, but at some point we're gonna," Trump said on Wednesday.
Oil prices initially eased on hopes of a deal that would allow shipping to resume but later inched upward.
Brent crude, the international standard, was about $US80 per barrel on Wednesday - still well below the levels it hit at the height of the conflict.
Energy prices have fluctuated wildly during the conflict as shipments through the key route for around 20 per cent of the world's oil and liquefied natural gas (LNG) have been severely disrupted.
Since the beginning of the war, traffic through the Strait of Hormuz has dwindled. The US and Iran reached a deal to open the crucial waterway in June, only for attacks to resume.
with AP and DPA
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