Meanwhile, a delegation from Qatar travelled to Iran on Friday for talks aimed at defusing tensions and easing navigation through the Strait of Hormuz.
Trump wrote in a post on Truth Social earlier on Friday: “The Islamic Republic of Iran has asked us to continue ‘talks.’
“We have agreed to do so, but the United States has stated to them, in no uncertain terms, that the Cease Fire is OVER!”
In the early hours of Saturday, Trump also responded to reports that Iran had plans to assassinate him.
Writing on Truth Social, he said that the US army would “completely decimate and destroy all areas” of the country in retaliation to such an attack.
The Wall Street Journal and other US media reported this week that Israel had shared intelligence with Washington that Iran had recently devised a plan to assassinate the US president.
However, Trump denied that Tehran had made a fresh plan or that Israel was the source of any intelligence. He told the New York Post in an interview that he had been “No. 1 [on Iran’s kill list] for a long time”.
There were also open calls for Trump’s death at the funeral of Iran’s late Supreme leader Ayatollah Ali Khamenei.
Khamenei, who was buried this week, was killed in an Israeli strike on his residence in Tehran on 28 February, the first day of Iran’s war with the US and Israel.
On Saturday, Iran’s supreme leader Mojtaba Khamenei released a written message that said vengeance for his father’s killing was “inevitable”.
In the first statement since his father’s funeral, he said the “matter depends neither on my personal existence nor on that of other officials. Whether we are present or not, it will come to pass.”
No fresh attacks were reported on Friday after fighting broke out in the Gulf region earlier this week, marking the worst exchange of fire between the US and Iran since the two nations signed an interim deal in June.
Three ships were struck while using a US-recommended route through Omani waters. Iran has repeatedly said the only “safe” passage is a separate route through its waters.
Overall progress came last month when the US and Iran agreed on a 14-point memorandum of understanding, which was aimed at extending a ceasefire and ending conflict “on all fronts”.
As part of the agreement Iran and Oman must hold talks “to define the future administration and maritime services” in the strait with other Gulf states.
During the conflict, Iran sought to assert its sovereignty over the strait, including by establishing the “Persian Gulf Strait Authority”, which it said would manage “safe passage permits”.
Iran’s Fars news agency has reported that under the new deal with the US the strait would ultimately be managed by Iran in co-ordination with Oman, including possible “service fees” for ships to transit the waterway.