Boris Johnson said he "did not envisage" any escalation in the tensions between the US and Iran and urged Donald Trump to produce a replacement for the nuclear deal.
The UK, along with European allies, has remained part of the troubled nuclear deal despite the US president's opposition.
The prime minister said one of the problems the American leader had with the agreement was that it was negotiated by his predecessor, Barack Obama, so putting in a replacement "Trump deal" would be a way out of the current crisis.
The US has pulled out of the Joint Comprehensive Plan of Action (JCPOA), which eased sanctions in return for Iran agreeing to restrictions on its activity in an effort to ensure Tehran did not acquire a nuclear weapon.
The UK and European allies have publicly remained committed to the approach, despite breaches by Iran and the difficulties caused by Mr Trump's decision to walk away.
Mr Johnson told BBC Breakfast: "If we are going to get rid of it then we need a replacement.
"The problem with the JCPOA - this is the crucial thing, it's why there is this tension - the problem with the agreement is that from the American perspective it's a flawed agreement, it expires, plus it was negotiated by President Obama.
"From their point of view it has many, many faults.
"If we are going to get rid of it, let's replace it and let's replace it with the Trump deal. That's what we need to see.
"I think that would be a great way forward.
"President Trump is a great deal-maker - by his own account and many others. Let's work together to replace the JCPOA and get the Trump deal instead."
The prime minister said he "did not envisage" any escalation in the tensions between the US and Iran, adding: "Let's dial this thing down."
In an interview with BBC Breakfast, the Prime Minister said he was "glad" Iran had acknowledged it made a "terrible mistake" in shooting down the Ukrainian passenger plane last week.
His comments came shortly after Iran's president announced arrests have been made over the incident and called for a special court to investigate.
Mr Johnson said the next step was to "repatriate in a dignified way" the Britons who had lost their lives in the tragedy.
He said: "I'm glad the Iranians have accepted responsibility and identified it as an appalling mistake and it does appear that it was a mistake.
"It is very important that the bodies are repatriated in a dignified way and that the families are allowed to grieve and to have closure.
"Clearly, as President Rouhani has said, Iran made a terrible mistake. It is good they have apologised.
"The most important thing now is that tensions in the region calm down."
The prime minister also defended his role during the tensions with Iran, after facing criticism for not returning from his holiday immediately.
"I was not in this country but I worked very hard, as you can imagine, in making sure there was a European response," he said.
Mr Johnson said Britain had played its "traditional role" in serving as the bridge between the "European powers and the United States".
He said there was no need for Britain to have been informed before the attack: "This was not our operation. There was no reason for us to be told."
Meanwhile Iran's judiciary said on Tuesday arrests have been made over the accidental shootdown.
The announcement came shortly after Iran's president called for a special court to be set up to probe the downing last week of the plane by Iranian forces.
Judiciary spokesman Gholamhossein Esmaili was quoted by Iranian state media saying extensive investigations have taken place and some individuals are arrested.
He did not say how many individuals have been detained or name them.
Iran, which initially dismissed allegations that a missile had brought down the jetliner, acknowledged three days after Wednesday's downing and in the face of mounting evidence that its Revolutionary Guard had shot down the Ukrainian plane by mistake.
"The judiciary should form a special court with a ranking judge and dozens of experts," President Hassan Rouhani said in a speech televised in Iran.
"This is not an ordinary case. The entire the world will be watching this court."
Rouhani called the incident "a painful and unforgivable" mistake and promised that his administration would pursue the case by all means.
"The responsibility falls on more than just one person," he said, adding that those found culpable should be punished.
Rouhani called the government's admission that Iranian forces shot down the plane the "first good step".
The plane, en route to the Ukrainian capital of Kyiv, was carrying 167 passengers and nine crew members from several countries, including 82 Iranians, 57 Canadians - including many Iranians with dual citizenship - and 11 Ukrainians, according to officials.
There were several children among the passengers, including an infant.
Iran shot down the plane when it was bracing for possible US retaliation for a ballistic missile attack on two military bases housing US troops in Iraq.
No one was hurt in that attack, which was carried out to avenge the stunning killing of Revolutionary Guard Gen. Qassem Soleimani in an American airstrike in Baghdad.
The shootdown and the lack of transparency around it has reignited anger in Iran at the country leadership, with protesters taking to the streets.
Online videos appeared to show security forces firing live ammunition and tear gas to disperse protests in the streets.