The US has said it hit Iranian military sites over the weekend while Tehran said it responded by targeting a US base, marking the third escalation in a week around the Strait of Hormuz.
US Central Command (Centcom) said it launched "self-defence strikes" in response to "aggressive Iranian actions", which included a US drone being shot down over international waters.
Iran's Islamic Revolutionary Guard Corps (IRGC) said it had targeted an air base used by US forces for an attack on Iran.
Centcom said two Iranian ballistic missiles targeting US forces in Kuwait had been intercepted and "no American personnel were harmed".
Kuwait earlier said its forces had confronted "hostile" missiles and drones.
US President Donald Trump urged his critics to "sit back and relax" in a post on Truth Social early on Monday, saying it would "all work out well in the end". He said Iran "really wants to make a deal, and it will be a good one for the USA".
The strikes mark the latest exchange between the two sides after negotiations on a deal to end their months-long war failed to advance over the weekend, with US media reporting Trump had requested changes to its terms.
The changes are related to the Strait of Hormuz shipping channel and the removal of highly enriched uranium from Iran, the BBC's US news partner CBS News reported. The White House did not respond to a request for comment.
On Monday, Iran's foreign ministry spokesman said the US was "constantly changing its views and putting forward new or contradictory demands", which he said would naturally "prolong negotiations".
The country's chief negotiator had said on Sunday that Tehran would not agree to any deal unless Iranian rights were fully secured.
The US military said it had on Saturday and Sunday conducted "self-defence strikes on Iranian radar and command and control sites for drones" in the city of Goruk, near Iran's southern coast, and Qeshm, an island in the Strait of Hormuz.
In a post on X, Centcom said US fighters struck the Iranian military's air defences, a ground control station and two drones that it said "posed a clear threat to ships transiting through regional waters". No American personnel were injured in the attacks, it said.
Iran's foreign ministry spokesman said the strikes had been a violation of the ceasefire.
The IRGC said it targeted the base which it claimed the US had used to strike a communications tower on Sirik Island in the Gulf, around 40 miles (65km) from Iran's southern coastline.
Iran's military added that its response would be "completely different" if US aggression was "repeated", according to IRGC remarks reported by Iran's semi-official Fars news agency.
Kuwait's military said early on Monday it was "confronting hostile missile and drone attacks", with state news agency KUNA reporting air raid sirens sounded across the country.
Its foreign ministry later released a statement condemning "in the strongest terms... the heinous and repeated Iranian attacks, which represent a dangerous escalation and a direct assault" on Kuwait.
It said such attacks "undermine" efforts to deescalate tensions in the region and said the country reserved the right to "take whatever measures are necessary" to defend itself.
Tehran targeted an air base in Kuwait last week in response to earlier US air strikes, which it said were conducted to prevent Iranian boats and missile strikes from laying mines around the shipping channel.
While a ceasefire came into effect on 8 April, Trump has repeatedly suggested the US and Iran are close to a permanent deal and that negotiations are progressing, but so far no formal agreement has been reached.
Trump and senior aides met on Friday to make a "final determination" about a framework for extending the ceasefire, but the meeting concluded without clarity on the next steps before reports later emerged that the president had requested changes to the text.
The latest terms include a 60-day cessation of violence and a call to reopen the Strait of Hormuz, CBS News reported - the shipping lane through which approximately one-fifth of global oil and liquefied natural gas (LNG) shipments usually pass, whose effective closure has sent global oil prices soaring.
It also reportedly includes a framework to reopen negotiations on Iran's nuclear programme - which Tehran has long maintained is peaceful - although Iran's foreign ministry spokesman Esmail Baghaei denied this had been on the table on Monday.
"No negotiations have taken place on the details of the nuclear file. At this stage, our priority is ending the war," Baghaei told a press briefing.
He added that an end to the conflict in Lebanon remained an "essential condition" for any deal and that Washington and Tehran had not yet reached "a final conclusion".
Iran's foreign minister Abbas Araghchi echoed this in an X post on Monday, writing: "The ceasefire between Iran and the US is unequivocally a ceasefire on all fronts, including in Lebanon."
"The US and Israel are responsible for the consequences of any violation," he added.
Lebanon was drawn into the war between the US, Israel and Iran on 2 March.
The Iran-backed group Hezbollah launched rockets into Israel in retaliation for an Israeli strike that killed Iran's Supreme Leader Ali Khamenei, to which Israel responded with an air campaign across Lebanon and a ground invasion.
