Russia and Germany have urged countries in the Middle East to show restraint and Israel says it is preparing to "meet all its security needs" in a region on edge over an Iranian threat to strike Israel.
The German airline Lufthansa extended a suspension of its flights to the Iranian capital Tehran and Russia warned against travel to the Middle East.
Iran has vowed revenge for the April 1 air strike on its embassy compound in Damascus that killed a top Iranian general and six other Iranian military officers, ratcheting up tension in a region already strained by the Gaza war.
Israel has not claimed responsibility for the attack but Iranian Supreme Leader Ayatollah Ali Khamenei said on Wednesday Israel "must be punished and it shall be," saying it was tantamount to an attack on Iranian soil.
The "imperative for Iran to punish this rogue regime" might have been avoided had the United Nations Security Council condemned the strike and brought the perpetrators to justice, Iran's mission to the United Nations said on Thursday.
Prime Minister Benjamin Netanyahu said Israel was keeping up its war in Gaza but making security preparations elsewhere.
"Whoever harms us, we will harm them. We are prepared to meet all of the security needs of the state of Israel, both defensively and offensively," he said in comments released following a visit to an air force base.
Conflict has spread across the Middle East since the eruption of the Gaza war, with Iran-aligned groups declaring support for the Palestinians waging attacks from Lebanon, Yemen and Iraq.
Iran has avoided direct confrontation with Israel or the United States, while declaring support for its allies.
German Foreign Minister Annalena Baerbock called on her Iranian counterpart Hossein Amirabdollahian to urge "maximum restraint" to avoid further escalation.
Russia's foreign ministry told citizens they should not travel to the Middle East, especially to Israel, Lebanon and the Palestinian territories.
"Right now it's very important for everyone to maintain restraint so as not to lead to a complete destabilisation of the situation in the region, which doesn't exactly shine with stability and predictability," Kremlin spokesman Dmitry Peskov told a news briefing.
US President Joe Biden said on Wednesday Iran was threatening to launch a "significant attack in Israel" and that he had told Netanyahu that "our commitment to Israel security against these threats from Iran and its proxies is iron-clad".
Iran is the third-largest oil producer in the OPEC group and oil prices stayed near six-month highs on Thursday.
Late on Wednesday, an Iranian news agency published an Arabic report on the X platform saying the air space over Tehran had been closed for military drills but then removed the report and denied it had issued such news.
Lufthansa said it would probably not fly to Tehran before April 13.
Austrian Airlines said it was still planning to fly on Thursday but was adjusting timings to avoid crew having to disembark for an overnight layover.
Iranian air space is also a key overflight route for Emirates' and Qatar Airways' flights to Europe and North America.
Emirates, Qatar Airways, Turkish Airlines, Aeroflot and Air Arabia, among the airlines that fly to Tehran, did not immediately respond to requests for comment.
US Secretary of State Antony Blinken spoke to the Turkish, Chinese and Saudi Arabian foreign ministers in the past 24 hours to make clear that escalation in the Middle East is not in anyone's interest, the State Department said on Thursday.
