Remainer James Whale said only people who were on higher wages had a "stake in society".
Britain voted by 52 per cent to 48 per cent in favour of Brexit in the historic referendum back in 2016, where young, old, rich and poor turned out to have their say.
But the talkRADIO presenter said: "There are lots of very thick and stupid people around… there are a lot of people who shouldn't have the vote."
And he went on: "Let's stop worrying about offending people and actually say, not everyone weighs everything up and thinks about things seriously.
"Some people just think ‘oh I heard that I might do this’…
"That is not good for democracy is it?
Radio host Ash Gould added: "Ash doesn’t think that anybody who doesn’t earn over £50,000 a year should be allowed to vote."
Tory MP Mark Francois said he would have to "agree to disagree" on that point, and believed that everyone should be allowed to cast their ballots regardless of income.
He said: "You're saying essentially that the British public are a bit thick?"
But one furious Brit wrote on Twitter: "Now we see how Remoaners would like to destroy our rights!"
And another added: "Money doesn't make a person intelligent, look at all the stupid celebrities out there."
Mr Francois went on to to argue that the first referendum was a clear result, and should not be re-run under any circumstances.
"The decision was to leave, over a million votes.
"Even Diane Abbott knows that's a big number," he joked.
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