Trump's comments came after Iranian state TV reported it had obtained an unofficial draft of an agreement that would restore commercial shipping through the strategic waterway to prewar levels within a month, with Iran and Oman jointly managing traffic.
That framework also would have the United States lift its blockade of Iranian ports and withdraw military forces from Iran's vicinity.
Trump said no single country would have control over the waterway, and appeared to threaten Oman, a country with which the US has decades-long military and economic ties.
"Nobody's going to control (the strait)," Trump said at a cabinet meeting attended by reporters on Wednesday, US time.
"It's international waters and Oman will behave just like everybody else or we'll have to blow them up. They understand that, they'll be fine."
The White House and Oman's embassy in Washington did not immediately respond to a request for comment.
Trump's statement indicated the two countries remain far apart as they seek an initial deal to end the war, which has killed thousands and sent global energy prices sharply higher since it began with US and Israeli strikes on February 28.
Trump has repeatedly said a deal is close at hand since a ceasefire took effect in early April.
Sticking points in talks seeking to end the three-month-long conflict include reopening the strait, through which a fifth of the world's oil and liquefied natural gas travelled before the war, and the issue of the dismantling of Iran's nuclear capacity.
"We'll watch over it, but nobody's going to control it - that's part of the negotiation that we have," Trump said of the strait.
Trump has also asked Saudi Arabia, Qatar, Pakistan, Turkey, Egypt and Jordan to join the Abraham Accords normalising relations with Israel as part of a deal to end the war, which they have declined to do.
He brought up the issue again at the cabinet meeting. "I'm not sure we should make the deal if they don't sign, if you want to know the truth," Trump said.
Iranian state TV said the draft deal would also have the United States withdraw military forces from the immediate vicinity, though it said the issue of US troops in the region needed further discussion.
The White House dismissed the report as a "complete fabrication". Tehran did not comment.
Oil prices fell more than five per cent after the Iranian television report, before regaining about a fifth of that fall.
The Iranian TV report did not mention Iran's nuclear program, which the US wants disbanded. Iranian sources have said talks on the nuclear issue will come in a second round of negotiations - something that may not be acceptable to some of Trump's closest supporters. Iran says its nuclear program is for peaceful purposes only.
"The bottom line is Iran's never going to have a nuclear weapon," US Secretary of State Marco Rubio said at the cabinet meeting.
Iran's Islamic Revolutionary Guards Navy said on Wednesday 23 ships including oil tankers, container ships and other commercial vessels passed through Hormuz with its permission in the previous 24 hours, a fraction of the daily 125 to 140 vessels passing through before the conflict.
The war has also created political problems for Trump at home. US polls show the war is deeply unpopular with the public, less than six months before midterm elections that will determine whether Trump's Republican Party retains control of the Senate and House of Representatives.
with AP
