Hopes that the deadly conflict can be contained helped most equities rise, while the US president's earlier claim that the Islamic republic wanted to make a nuclear deal also provided a little optimism.

After Friday's surge sparked by Israel's attacks on its regional foe, crude ticked more than one percent lower Monday as traders bet that the battle would not spread throughout the Middle East and key oil sites were mostly left untouched.

But prices edged back up after Trump took to social media calling for the evacuation of the Iranian capital, which is home to nearly 10 million people.

"Iran should have signed the 'deal' I told them to sign," he said, referring to nuclear talks that were taking place. 

"What a shame, and waste of human life. Simply stated, IRAN CAN NOT HAVE A NUCLEAR WEAPON. I said it over and over again! Everyone should immediately evacuate Tehran!"

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Oil prices spiked around 2 percent Tuesday before paring some of those gains, but the comments kept investors on edge amid warnings that an escalation of the crisis could send the commodity soaring again.

Meanwhile, the aircraft carrier USS Nimitz left Southeast Asia on Monday after cancelling a Vietnam visit, with the Pentagon announcing it was sending "additional capabilities" to the Middle East.

Prime Minister Benjamin Netanyahu insisted Israel's campaign was "changing the face of the Middle East".

Trump has maintained that Washington has "nothing to do" with its ally's campaign, but Iran's foreign minister said Monday that the US leader could halt the attacks with "one phone call".

Traders had been a little more upbeat after the US president -- who is in Canada for the G7 summit -- had said Iran wanted to make a deal, saying "as soon as I leave here, we're going to be doing something". 

He later left the gathering in the Rockies, telling reporters: "I have to be back as soon as I can. I wish I could stay for tomorrow, but they understand, this is big stuff."

Tehran had signaled a desire to de-escalate and resume nuclear talks with Washington as the United States did not join conflict, according to the Wall Street Journal. 

Stocks mostly rose in Asian trade, with Tokyo, Sydney, Seoul, Singapore and Taipei leading gains, though Shanghai and Hong Kong struggled.

"Risk assets are enjoying a positive start to the new week amid signs the Israel-Iran war remains limited to the two countries without signs of a possible escalation into a wider conflict," said Rodrigo Catril at National Australia Bank.

"Iran is reportedly seeking de-escalation talks, but Israel is not showing signs of slowing down."

The gains followed a positive lead from Wall Street, where traders are keeping tabs on the G7 summit world leaders pushed back against Trump's trade war, arguing it posed a risk to global economic stability.

Leaders from Britain, Canada, Italy, Japan, Germany and France called on the president to reverse course on his plans to impose even steeper tariffs on countries across the globe next month.

Also in view are central bank decisions this week, with the Bank of Japan due to make its latest decision on interest rates later in the day.

Officials are expected to hold interest rates steady but tweak their bond purchase policy.

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