Tensions between the US and Iran have increased in recent weeks after the downing of an American drone and claims by Washington that Tehran was behind attacks on oil tankers in the Persian Gulf.
The Foreign Secretary's comments came as Foreign Office minister Andrew Murrison held talks with the Tehran government at the weekend where he said he was "clear" about the UK's concerns over Iran's activities.
While campaigning in Scotland for the Tory leadership Mr Hunt said Britain would weigh up military intervention in Iran on a "case-by-case basis".
"We will stand by the United States as our strongest ally but of course we have to consider any requests for military support on a case-by-case basis," he told the Daily Mail.
"We do strongly believe that the solution is for Iran to stop its destabilising activity throughout the Middle East and we are very concerned about the sabotaging of tankers that has happened recently, which is almost certainly Iran, and we're constantly in touch with the United States.
"We want to de-escalate the situation but we are of course extremely worried."
Mr Hunt, who is battling to become the next prime minister, earlier said he has been speaking to counterparts "regularly" about the Iran crisis which he called "extremely serious".
US secretary of state Mike Pompeo said on Sunday that he was seeking to build a "global coalition" against Iran, branding Tehran "the world's largest state sponsor of terror".
Before flying to Saudi Arabia and the UAE for talks, Mr Pompeo said the US was prepared to negotiate with Iran without preconditions when the country is "ready to truly engage with us".
He added that he wants to build an alliance "not only throughout the Gulf states, but in Asia and in Europe that understands this challenge as is prepared to push back against the world's largest state sponsor of terror".
"We will continue to make sure it's understood that this effort that we've engaged in - to deny Iran the resources to foment terror, to build out their nuclear weapons system, to build out their missile programme - we are going to deny them the resources they need to do that," Mr Pompeo told reporters at Andrews Air Force Base.
Earlier Dr Murrison said his Tehran visit had provided an important opportunity for "open, frank and constructive engagement with the Iranian Government".
"I was clear about the UK's long-held concerns over Iran's activities in the region," he said.
"And I was clear that the UK will continue to play its full part alongside international partners to find diplomatic solutions to reduce the current tensions.
"I reiterated the UK's determination to maintain the nuclear deal which is in our shared security interests.
"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."
The trip was announced after Donald Trump said the US was "cocked and loaded" to retaliate against Iran for downing an American drone.
The strikes were reportedly called off 10 minutes before they were to be carried out on Thursday after Mr Trump was told 150 people could die.
On Saturday, a spokesman for Iran's armed forces warned that a military strike would "set the region ablaze and burn up the US, its interests, and its allies".
Mr Trump later tweeted that the US will impose "major additional sanctions" on Iran on Monday after opting to halt military action "from going forward at this time".
Mr Trump pulled out of a long-term deal struck with Iran in 2015 to limit its sensitive nuclear activities and allow in international inspectors in return for the lifting of economic sanctions.
The deal was signed by his predecessor Barack Obama along with the UK, France, China, Russia and Germany, which still support the agreement.
But Mr Trump declared it a "terrible" deal and imposed more sanctions.