Iran and Israel on Monday attacked each other's territory for the first time since a shaky ceasefire put five weeks of war on hold, sparking fears the escalation could spark a new full-scale conflict.

Iran fired a barrage of missiles at Israel overnight and Israel responded by targeting military sites in the Islamic republic, with Israeli Prime Minister Benjamin Netanyahu apparently defying calls by his ally President Donald Trump for restraint.

Tehran's strikes followed attacks by Israel against targets of the Iran-backed Lebanese Shia group Hezbollah in the southern suburbs of Beirut. Iran had repeatedly warned it would strike Israel if the Lebanese capital was targeted.

Oil prices surged on worries that war could break out again, with hopes now punctured of a rapid end to the standoff that has seen shipping limited through the key Strait of Hormuz trade bottleneck amid fears of global energy and goods shortages.

The strikes also came at a critical moment with diplomatic efforts to end the conflict involving mediator Pakistan on a knife-edge.

Iranian foreign ministry spokesman Esmaeil Baqaei warned at a press conference in Tehran attended by AFP that is was "perfectly natural that the diplomatic process initiated to put an end to this imposed war would be affected."

But he added: "Diplomatic consultations are naturally continuing in all circumstances."

As he was speaking at the foreign ministry, a huge explosion shook the building, followed by repeated explosions believed to be from air defence systems, the AFP reporter said.

Local media in Iran that a "hostile drone" was shot down over Tehran by air defences.

- 'Prior coordination' -

No casualties have been reported so far in either Israel or Iran.

The Israeli military said it struck and dismantled Iranian defence systems deployed across several areas in the country.

AFP journalists in Jerusalem and the West Bank city of Ramallah heard a series of explosions and the Israeli army said it worked to intercept a new wave of Iranian missiles.

Iran's Revolutionary Guards said they had struck Israel's Nevatim and Tel Nof air bases and had also targeted a petrochemical facility in Israel in retaliation for an attack on a similar site in southwestern Iran.

The Guards warned that Israel "has initiated a dangerous game, the scope of which will encompass all energy-related targets in the region".

Foreign ministry spokesman Baqaei blamed the US for the flare-up, saying "no-one believes" Israel would carry out any action "without prior coordination and cooperation with the United States".

- 'I call the shots' -

Yemen's Iran-backed Houthi rebels meanwhile announced a missile attack on Israel on Monday, the first since early April, and declared a ban on Israeli shipping in the Red Sea.

Trump called for calm from both Netanyahu and the Islamic republic, but Israel accused Tehran of making a "grave mistake".

"I call the shots. I call all the shots. He doesn't call the shots," Trump said in an interview with the Financial Times on Sunday, referring to Netanyahu.

In an interview with Fox News, Trump said: "What I would suggest to Iran: You've shot your missiles, that's enough, get back to the table and make a deal."

There were some weekend signs of ongoing diplomatic efforts, with Pakistan Interior Minister Mohsin Naqvi visiting Tehran to deliver what he said was a "special letter" to Iran's supreme leader Mojtaba Khamenei, according to Iranian state television.

He has since travelled back to Pakistan, an official Pakistani source said on Monday.

- 'Does not need escalation' -

The European Union's top diplomat Kaja Kallas called on both sides to "sit down to a negotiation table and agree", adding that "the region does not need an escalation."

China also called on the two sides to refrain from fighting, with foreign ministry spokesman Lin Jian saying that "resuming hostilities is not in any party's interest".

Tehran has insisted any deal to permanently end the war must also halt the parallel conflict in Lebanon, where Israel was pursuing a campaign against Hezbollah.