Speaking at a briefing with reporters, the US official added that there will also be a signing ceremony on Friday.
"You will see significant increase in traffic in the Strait of Hormuz, actually starting already, and that will ramp up slowly over time," the US official, speaking on the condition of anonymity, said.
"We probably won't return to normal in two weeks but we will see a significant increase in strait traffic," the official said.
The US and Iran said they had agreed terms to end their war and reopen the strait, news that brought relief to markets, although the pact may hinge on an end to hostilities in Lebanon and defers talks on Iran's nuclear program.
While still a framework, the deal marked the biggest breakthrough toward resolving the conflict that has killed thousands and upended energy markets since it began with joint US-Israeli strikes on Iran in February.
Trump on Monday said details of the agreement would be released sometime after a formal signing on Friday, adding that the Strait of Hormuz would also be fully open.
Speaking alongside French President Emmanuel Macron ahead of this week's G7 meeting, Trump said he did not know if he would attend the Friday ceremony expected in Genev, but that Vance would be there.
"The deal's all signed. And the strait is already partially opened, as you know," Trump told reporters shortly after arriving in Evian, France.
"On Friday, it'll be completely open."
He added: "I think a lot of great things are going to happen in the Middle East right now and, very importantly, the oil is plummeting down and the stock market is shooting up like a rocket today."
"The main thing is that Iran will not have a nuclear weapon, they fully agreed to that with strong policing powers, and they won't have a nuclear weapon, which is what it was all about, because they probably would have used it if they had it," Trump said.
Vance earlier on Monday said the agreement had been signed digitally on Sunday and that no funds were released.
Asked when the text of the memorandum of understanding would be made public, Trump said: "Probably pretty soon. I would say after sometime after Friday ... I think sometime in the very near future."
Trump said any sanctions relief for Iran was "really a behavioural thing. If they do what they're supposed to do, that starts taking effect".
with PA
