The Republican candidate tells donors the Middle East will “remain an unsolved problem… and we kick the ball down the field”.
The video is from the same event as a clip released on Monday, in which Mitt Romney says almost half of Americans “believe that they are victims”.
It comes less than two months before the presidential election.
The new footage was posted on Tuesday on the website of the liberal investigative magazine Mother Jones, which said it was taken at a fundraiser in Boca Raton, Florida.
Mitt Romney is shown saying that Palestinians are “committed to the destruction and elimination of Israel”.
“The Palestinians have no interest whatsoever in establishing peace,” he says, adding that “the pathway to peace is almost unthinkable to accomplish“.
In another clip, the former Massachusetts governor is shown discussing Iran’s nuclear programme, and warning that America itself could come under attack.
“If I were Iran – a crazed fanatic, I’d say let’s get a little fissile material to Hezbollah, have them carry it to Chicago or some other place, and then if anything goes wrong, or America starts acting up, we’ll just say, <<Guess what? Unless you stand down, why, we’re going to let off a dirty bomb>>.”
On Monday, Mother Jones posted another clip from the same fundraiser in which Mitt Romney disparages those who would vote for his Democratic rival, President Barack Obama.
The Republican candidate is shown saying that the 47% of Americans who back the president do not pay income tax and would never vote for Mitt Romney.
“There are 47% who are with him [Barack Obama], who are dependent upon government, who believe that they are victims, who believe the government has a responsibility to care for them, who believe that they are entitled to health care, to food, to housing, to you-name-it.”
Mitt Romney said in the video that his role “is not to worry about those people. I’ll never convince them they should take personal responsibility and care for their lives.”
In a late-night appearance on Monday, the Republican candidate said those remarks were not “elegantly stated”, but did not retract them.
The Obama campaign was quick to pounce.
“It’s hard to serve as president for all Americans when you’ve disdainfully written off half the nation,” Jim Messina, Barack Obama’s campaign manager, said in a statement.
The video clips provided to Mother Jones are said to have been filmed at a $50,000 a head fundraiser at some point after Mitt Romney became the presumptive Republican nominee.
The pictures are blurred out with the exception of his face, and no attendees are visible.
Also on Monday, Mitt Romney’s campaign unveiled a significant reworking of its strategy less than two months before election day.
Until now his message to voters has largely consisted of repeated attacks on President Barack Obama’s economic record.
Campaign advisers told the US media on Monday that Mitt Romney would speak more specifically about his budget plans and tax policy.
The videos bookended a difficult 24 hours for the Romney campaign. On Sunday, a lengthy Politico story detailed apparent divisions and indecision within the campaign.
A series of opinion polls at the end of last week showed Mitt Romney trailing Barack Obama both nationally and in several swing states.
The two men square off in the election on 6 November, but early voting begins soon in several states.
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