Current UKIP leader Henry Bolton said the party did not support a new referendum because it would "undermine the fabric of our democratic principles".
Pro-Brexit politician Nigel Farage suggested a second vote could help "silence" pro-Remain voters seeking to overturn the result of the EU Referendum.
"What is for certain is that the Cleggs, the Blairs, the Adonises will never, ever, ever give up". They will go on whinging and whining and moaning throughout this process. "The percentage that would vote to leave would be very much bigger than last time and we may just finish the whole thing off and Blair could disappear into total obscurity".
On Thursday, a ComRes poll for the Daily Mirror newspaper of 1,049 adults showed that although more people think there should not be a second referendum (51 to 43 per cent), if there were to be a re-run, voters say they would opt to stay in the European Union by 55 to 45 per cent.
Many of them reacted quickly in support of Farage's proposal, expressing confidence about winning the second time around.
"No one said Brexit would be straightforward, everyone said there would be problems initially".
Sittingbourne and Sheppey MP Gordon Henderson said: "I have always been a great supporter and admirer of Nigel Farage but I am becoming slowly less warm now that he has become a caricature of himself to get attention". This hasn't happened. It hasn't happened because people are becoming more anxious about the consequences.
"I don't agree with the decision and it's not the way I would vote but I think that [having a second referendum] is a ridiculous idea".
Europhiles seized on the remarks, which Mr Farage later backtracked on, hailing them as a "game changer" in their crusade to keep Britain in the EU.
Prime Minister Theresa May clinched a divorce deal last month that paves the way for arduous talks on future trade ties ahead of the United Kingdom's exit which is due on March 29, 2019.