A US service member who has been missing since Iran shot down a fighter jet has been rescued, two US officials said on Sunday.

The officials spoke on the condition of anonymity ahead of an official announcement.

It comes after a frantic search-and-rescue operation.

The crew member has been missing since Friday, when Iran downed a US F-15E Strike Eagle. A second crew member was rescued earlier.

"The United States Military pulled off one of the most daring Search and Rescue Operations in US History, for one of our incredible Crew Member Officers, who also happens to be a highly respected Colonel, and who I am thrilled to let you know is now SAFE and SOUND," Trump posted on Truth Social.

Trump wrote that the aviator is injured but "will be just fine," adding that he took refuge "on the treacherous mountains of Iran."

As US forces converged on the downed airman, a firefight erupted, a former senior military official briefed on the operation said.

Trump added that the rescue involved "dozens of aircraft" and that US had been monitoring his location "24 hours a day, and diligently planning for his rescue."

Earlier, Trump warned Iran to open the crucial Strait of Hormuz by his Monday deadline and Tehran called his threat "unbalanced and foolish".

The war began with joint US-Israel strikes on February 28 and has killed thousands, shaken global markets, cut off key shipping routes and spiked fuel prices. Both sides have threatened, and hit, civilian targets, bringing warnings of possible war crimes.

The fighter jet was the first US aircraft to have crashed in Iranian territory since the conflict in late February.

Trump said last week that the US had "decimated" Iran and would finish the war "very fast."

Two days later, Iran shot down two US military planes, showing the ongoing perils of the bombing campaign and the ability of a degraded Iranian military to continue to hit back.

The other jet to go down was a US A-10 attack aircraft. Neither the status of the crew nor exactly where it crashed was immediately known.

A frantic US search-and-rescue operation unfolded after the crash of the F-15E jet on Friday, focusing on a mountainous region in Iran's southwestern province of Kohgiluyeh and Boyer-Ahmad.

Iran also promised a reward for anyone who turned in the "enemy pilot." Iran's joint military command on Saturday said that it also struck two US Black Hawk helicopters Friday, but The Associated Press couldn't independently verify that.