Jayne Adye, of campaign group Get Britain Out, said the only possibility would be if the "EU puts something more on the table quickly."
A UK parliamentary fightback against the result of the EU referendum gathered pace on Sunday, with pro-remain figures saying they would not "roll over and give up."
Some are urging a second referendum after Brexit negotiations have taken place.
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Former Tory cabinet minister Michael Heseltine has pointed to the overwhelming majority in the House of Commons against leaving the EU.
He said, "There is a majority of something like 350 in the House of Commons broadly in favour of the European relationship."
"There is no way you are going to get those people to say black is white and change their minds unless a) they know what the deal is and b) it has been supported either by an election or by another referendum," Heseltine told Sky News. "So there's a dramatic urgency to get on with the negotiations."
He called for a cross-party group of MPs to look at the options and "articulate the case for Britain rethinking the result of the referendum."
Tony Blair, the former UK Prime Minister, said nothing should be ruled out and Tim Farron, the Liberal Democrat leader, said his party would not "roll over and give up" over the issue, pledging to fight the next general election on a manifesto promise to take Britain back into the EU.
However, on Monday, Adye told this website, "Considering a great many people are pee'd off they didn't get Remain, it (a 2nd referendum) could get a debate.
"There are rallies in London and no doubt few will come from elsewhere. So those in London who voted no will press for this, but most people want to get on with life.
"As Angela Merkel, Jean-Claude Juncker and others are all pressing for an early exit I doubt it will end up with a referendum - unless of course the EU puts something more on the table quickly.
"They've already said they won't and have even withdrawn David Cameron's so-called negotiations.
"I also think it would be very expensive. So, all in all, I hope not."
Meanwhile, former UK Europe Minister Denis MacShane had told UKIP MEPs and pro-Brexit Tory MEPs to resign.
MacShane, who voted to remain in the EU, told this website, "Clearly if Ukip and pro-Brexit Tory MEPs were men of honour they would now be resigning to leave UK MEPs who are not anti-Europeans help the next government save what can be saved.
"Boris Johnson and Daniel Hannan, who campaigned obsessively on immigration and promised voters the government could control immigration, are now rowing back as they know UK will lose the single market if it tries to stop free movement.
"So they won the referendum on an out-and-out lie. Nigel Farage should now act with dignity and leave the institution (the European Parliament) he has constantly denigrated."