Ohio senator JD Vance, who is Mr Trump's vice presidential pick, made the comments in an address to the National Conservatism conference in Washington DC last week.
He said he had been discussing with a friend which would be "the first truly Islamist country that will get a nuclear weapon".
He said: "We were like maybe it's Iran, maybe Pakistan kind of counts, and then we sort of decided maybe it's actually the UK since Labour just took over.
"But to my Tory friends, I have to say, you guys have got to get a handle on this." Deputy Prime Minister Ms Rayner told how Mr Vance had said "quite a lot of fruity things in the past" and she "looked forward" to meeting him and Mr Trump if they won the US election in November.
She added: "I don't recognise that characterisation. We won votes across all different communities and we're interested in governing on behalf of Britain and also working with our international allies."
Tory MP Andrew Bowie, the shadow veterans minister, said he "absolutely" disagreed with the claim that Labour was creating an "Islamist country".
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