The latest flare-up, which sent oil prices higher, comes with the conflict stalemated in a shaky ceasefire and the Strait of Hormuz largely closed, more than three months after initial US and Israeli strikes on Iran.
Flights at Kuwait International Airport were suspended and diverted elsewhere until further notice after an Iranian drone and missile attack one of its terminals.
The attack caused injuries and severely damaged some airport facilities, state media reported.
Bahrain's army said it intercepted three missiles and several drones.
Earlier, the US Central Command said two Iranian missiles shot at Kuwait fell short or broke up in flight, while several ballistic missiles aimed at regional targets failed and three missiles heading for Bahrain were intercepted.
Since the conflict began, Iran has repeatedly attacked targets in the Gulf region home to US military bases.
Central Command said the US military also downed Iranian drones targeting civilian ships in regional waters and US forces in Kuwait, and carried out strikes on Qeshm Island near the Strait of Hormuz following attempted attacks by Iran.
Iran's state media said the Islamic Revolutionary Guard Corps (IRGC) attacked the headquarters of the US Fifth Fleet in Bahrain, as well as an air base and helicopters in an unspecified regional country.
It sent missiles and drones in response to what the IRGC described as a US attack on a communications tower south of Qeshm.
Central Command said all the attacks failed, however, and US forces stayed ready to repel "unwarranted Iranian aggression".
In late May, Iran and the United States said they had reached a tentative initial agreement to halt the war, but they have yet to finalise the deal.
Iranian media said Tehran has not communicated with Washington for several days, but US President Donald Trump said negotiations had not stopped.
"The conversations between us have been going on continuously, including four days ago, three days ago, two days ago, one day ago, and today," he said in a social media post.
Since mid-March, Trump has repeatedly said he is close to a deal to end the fighting and allow negotiators to tackle thorny issues, including the future of Iran's nuclear program.
Trump has said his top priority is to stop Iran from acquiring nuclear weapons.
Iran denies it is developing a nuclear bomb and says its atomic program is for peaceful purposes.
Tehran is seeking access to billions of dollars in oil revenues, waivers on crude exports, a lifting of a US blockade on its ports and continued leverage over the strait, a vital waterway for oil and gas traffic.
Iranian media said the IRGC's navy targeted a vessel it identified as the Panaya with missiles in response to what it said was a US attack on an Iranian tanker near Hormuz.
"Disrupting the security of the Strait of Hormuz will carry a heavy price for the US military," media cited the IRGC as saying.
US Secretary of State Marco Rubio told lawmakers on Tuesday the US would agree to sanctions relief only if Iran agreed to give up its nuclear activity.
"The war is over," Rubio declared during a sharp exchange with Democratic senator Cory Booker, who disagreed.
The war has killed thousands since it began on February 28, mainly in Iran and Lebanon, while also causing global economic pain by pushing up energy prices.
It also triggered fresh conflict between Israel and Lebanese militant group Hezbollah, with Israel pursuing its deepest incursion into Lebanon in 25 years.
On Tuesday, Israel kept up strikes on southern towns despite a US-mediated partial ceasefire unveiled on Monday.
