Israel, along with Arab countries in the Gulf, considers Iran to be their greatest threat, and Trump's last-minute about face appears to have raised questions about USA willingness to use force against the Islamic Republic.
Iran has continued its terror-related activities ever since it was first designated as a state sponsor of terrorism in 1984.
Tensions remained high after Mr Trump said on Friday that he aborted a military strike to retaliate for Iran's downing of an unmanned USA drone because it could have killed 150 people.
Foreign Minister Mohammad Javad Zarif Tweeted a map saying the US-made MQ9 Reaper drone - also widely used for carrying out military strikes - had entered his country's airspace on May 26.
A British Foreign Office official will visit Iran and call for "urgent de-escalation in the region and raise United Kingdom and worldwide concerns about Iran's regional conduct" during talks with Tehran's government.
However, the result must have been disappointing for the Iranian side, as Kharrazi said at the end of the meeting that Murrison "had the usual talking points", including discussions about a European payment mechanism to help Iran conduct its worldwide trade and collect its oil revenues regardless of USA sanctions.
Tehran is already feeling the effects of existing USA sanctions according to National security adviser John Bolton, who warned the Islamic regime on June 23 not to mistake "U.S. prudence and discretion for weakness".
"Neither Iran nor any other hostile actor should mistake USA prudence and discretion for weakness". The cyberattacks disabled Iranian Revolutionary Guard Corps computer systems that controlled its rocket and missile launchers, the officials said.
On Friday, Trump acknowledged that he had ordered a halt to a planned strike on three Iranian military sites that would have killed 150 people, reasoning that the death toll was disproportionate to Iran's shooting down of an unmanned drone earlier last week over the Strait of Hormuz.
U.S. Secretary of State Mike Pompeo repeated to reporters on Sunday that Washington wanted talks with Tehran.
The U.S. blames Iran for apparent attacks on six oil tankers in the Gulf of Oman in recent weeks, which Iran denies.
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Karen Bass (D-Calif.), the chair of the Congressional Black Caucus said that she hopes people will move on. Not the opposition, the enemy. "We don't talk to each other anymore".
Last year Trump withdrew the USA from a 2015 accord between Iran and world powers that curbed Tehran's nuclear programme in exchange for easing sanctions.
"When they agree to that, they're going to have a wealthy country".
I was clear that Iran must continue to meet its commitments under the deal in full - including the limits imposed on its low-enriched uranium stockpile.
A multinational accord reached by Tehran and world powers in 2015 sought to curb Iran's nuclear ambitions in exchange for sanctions relief.
But Trump left that agreement more than a year ago and has imposed a robust slate of punitive economic sanctions created to choke off Iranian oil sales and cripple its economy - which he now plans to expand.
"I reiterated the UK's assessment that Iran nearly certainly bears responsibility for recent attacks on tankers in the Gulf of Oman", Andrew Murrison, minister for the Middle East and North Africa, said in a statement.
"They know precisely how to find us", Pompeo said. "Until then, our diplomatic isolation and economic pressure campaign against the regime will intensify".
Iran has said it would respond firmly to any threat against it and warned on June 23 of the risks of a military confrontation.
Iran sounded more frustrated than before about Europe's cooperation following top-level meetings with the visiting British Minister of State for the Middle East Andrew Murrison on Sunday, June 23.
Netanyahu, a longtime opponent of the nuclear deal, has remained uncharacteristically quiet throughout the current crisis between the USA and Iran.
A longtime Iran hawk, Bolton emphasized that the USA reserved the right to attack at a later point.
The downing of the U.S. drone came after a series of attacks on tankers in the Persian Gulf and the Gulf of Oman, which Washington has blamed on Iran, exacerbated already-tense relations between the two countries.