Two witnesses reached by telephone in the Iranian capital said they heard two loud explosions.
Prime Minister Benjamin Netanyahu s office acknowledged that Israel had struck a radar site near Tehran in what it said was retaliation for Iranian missiles fired three-and-a-half hours after the ceasefire had been due to begin.
It said Israel had decided to refrain from further attacks following a call between Netanyahu and Trump, but did not explicitly say whether the strike on the radar site took place before or after they spoke.
Oil prices fell and share markets were lifted around the world at the prospect of an end to the biggest ever confrontation between the Middle East foes, lifting a threat of disruption to Gulf oil supplies.
Trump, en route to a NATO summit in the Netherlands, had rebuked Israel with an obscenity in an extraordinary outburst at an ally whose military campaign he had joined two days earlier.
"All planes will turn around and head home, while doing a friendly Plane Wave to Iran. Nobody will be hurt, the Ceasefire is in effect!" Trump said in a post on Truth Social.
That followed a post in which he had said: "Israel. Do not drop those bombs. If you do it it is a major violation. Bring your pilots home, now!"
Before departing the White House, Trump told reporters he was unhappy with both sides for violating the ceasefire, but particularly unhappy with Israel, which he said had "unloaded" shortly after agreeing the deal.
"I ve got to get Israel to calm down now," Trump said. Iran and Israel had been fighting "so long and so hard that they don t know what the fuck they re doing."
VIOLATIONS?
A reporter for Axios said Netanyahu had told Trump that Israel would scale back the bombing mission rather than cancel it.
Israeli Defence Minister Israel Katz had said earlier on Tuesday that he had ordered the military to mount major strikes on targets in Tehran in response to what he said were Iranian missiles fired in a "blatant violation" of the ceasefire.
Iran denied launching any missiles and said Israel s attacks had continued for an hour-and-a-half beyond the time the ceasefire was meant to start.
Despite the initial reports of violations, in both countries there was a palpable sense of relief that a path out of war had been charted, 12 days after Israel launched it with a surprise attack, and two days after Trump joined in with strikes on Iranian nuclear targets.
Read more: Iran is looking to restore its nuclear industry, Iran’s nuclear chief says
"We’re happy, very happy. Who mediated or how it happened doesn’t matter. The war is over. It never should have started in the first place," said Reza Sharifi, 38, heading back to Tehran from Rasht on the Caspian Sea where he had relocated with his family to escape strikes.
Arik Daimant, a software engineer in Tel Aviv, said: "Regrettably, it’s a bit too late for me and my family, because our house back here was totally destroyed in the recent bombings last Sunday. But as they say: better late than never , and I hope this ceasefire is a new beginning."
Speaking to reporters on board Air Force One, Trump said he did not want to see Iran s ruling system toppled.
"I don t want it. I d like to see everything calm down as quickly as possible. Regime change takes chaos and ideally we don t want to see so much chaos," he said.
"Iran s not going to have a nuclear weapon, by the way, I think it s the last thing on their mind right now," he said.
In the hours before the ceasefire took effect, four people were killed by Iranian missiles that struck a residential building in Beersheba in southern Israel, according to the Israeli ambulance service. Iranian officials said nine people were killed by a strike on a residential building in northern Iran.
NUCLEAR TARGETS
Israel launched a surprise attack on June 13, hitting Iranian nuclear sites and killing the top echelon of its military command in the worst threat faced by the Islamic Republic since war with Iraq in the 1980s.
During the campaign, Israel said it was prepared to topple Iran s clerical rulers if necessary to achieve its aims. Iran says its nuclear programme is for peaceful purposes and denies trying to develop nuclear weapons.
Iranian officials say hundreds of people have been killed in airstrikes. Full information about the extent of the damage cannot be confirmed independently, with media tightly controlled.
Retaliatory missile strikes have killed 28 people in Israel, the first time large numbers of Iranian missiles have penetrated its defences.
A senior White House official said Trump brokered the ceasefire deal with Netanyahu, and other administration officials were in touch with the Iranians. Qatar s Prime Minister Sheikh Mohammed bin Abdulrahman Al Thani secured Tehran s agreement during a call with Iranian officials, an official briefed on the negotiations told Reuters on Tuesday.
The ceasefire came the morning after Iran responded to the U.S. participation in the air strikes by firing a volley of missiles at the biggest U.S. military base in the Middle East, located in Qatar.
No one was hurt in that strike, with the Iranian retaliation appearing to have been calibrated carefully to allow de-escalation. Trump thanked Tehran for warning the United States in advance to avoid injuries.
