Ms Badenoch said the digital counter on Reform's website was "coded to tick up automatically" when it recorded membership on Boxing Day morning ticking past the 131,680 figure declared by the Conservative Party earlier this year.
She accused Reform UK of "fakery".
Demanding an apology from Ms Badenoch for what he said were "disgraceful accusations of fraud and dishonesty", Mr Farage vowed to take action in the next two days.
And when asked by journalists whether he would take legal action, he would not rule it out.
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Champion Humphries eases through to last 16
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Quad bike crash leaves girl with life-changing injuries
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Christmas travel chaos as fog causes mass flight delays
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Snow could sweep through UK as new year approaches
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