Sir Kim Darroch described the move as an act of "diplomatic vandalism", according to the Mail on Sunday.
The paper says the memo was written after the then Foreign Secretary Boris Johnson appealed to the US in 2018 to stick with the nuclear deal.
The latest leak came despite the Met Police warning against publication.
The first memos criticising President Trump's administration, which emerged a week ago, prompted a furious reaction from the US president and resulted in Sir Kim resigning from his role.
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What have we learnt from the latest leak?
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The Mail on Sunday reports that Sir Kim wrote to Mr Johnson informing him Republican President Trump appeared to be abandoning the nuclear deal for "personality reasons" - because the pact had been agreed by his Democrat predecessor, Barack Obama.
Under the deal, Iran agreed to limit its sensitive nuclear activities in return for the lifting of crippling economic sanctions.
However, President Trump said he did not think that the deal went far enough in curtailing Iran's nuclear ambitions and reinstated US sanctions after withdrawing from the deal in May 2018.
The British ambassador's memo is said to have highlighted splits amongst US presidential advisors; he wrote that the White House did not have a "day-to-day" strategy of how to proceed following withdrawal from the deal.
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What have we learnt from the latest leak?
The Mail on Sunday reports that Sir Kim wrote to Mr Johnson informing him Republican President Trump appeared to be abandoning the nuclear deal for "personality reasons" - because the pact had been agreed by his Democrat predecessor, Barack Obama.
Under the deal, Iran agreed to limit its sensitive nuclear activities in return for the lifting of crippling economic sanctions.
However, President Trump said he did not think that the deal went far enough in curtailing Iran's nuclear ambitions and reinstated US sanctions after withdrawing from the deal in May 2018.
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The British ambassador's memo is said to have highlighted splits amongst US presidential advisors; he wrote that the White House did not have a "day-to-day" strategy of how to proceed following withdrawal from the deal.
The Mail on Sunday said it was in the public interest to publish the memos.
The paper argued that the most recently published emails revealed "important information" on the UK's attempts to stop President Trump abandoning the Iran nuclear deal.
A spokesman said: "What could be more in the public interest than a better understanding of how this position was reached, which may have serious consequences for world peace?"
In response, a Foreign Office spokesman called it a "totally unacceptable leak" of "sensitive material" and called for the source of the leak to "face the consequences of their actions".
He added that it was "not news" that the UK and US differ in their approach to preventing Iran acquiring a nuclear weapon.
What was the fallout from the initial leak?
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The first leaked emails saw Sir Kim, the then UK ambassador, refer to the Trump administration as "clumsy and inept".
The US president responded by calling Sir Kim as "a very stupid guy", adding that he would no longer deal with him
Sir Kim stepped down as US ambassador on Wednesday, saying it was "impossible" for him to continue in the role.
Tory leadership candidate Boris Johnson faced strong criticism for failing to fully support the former ambassador during a TV debate last week.
The government have also launched an internal Whitehall inquiry into the leak.