The campaigns were launched after Boris Johnson suggested that members of the public could meet the £500,000 cost of the famous bell ringing at 11pm on 31 January, when the UK leaves the EU.
Almost 30 pages on the GoFundMe crowdfunding website are trying to raise the money, but the vast majority have not received a single donation.
The few that have received support have so far raised £551 – a tiny fraction of the amount needed.
Despite the muted response, Tory MP Mark Francois, who has led the campaign for Big Ben to ring on 31 January, predicted that the fundraising target would be met within two days.
The Brexiteer, who has pledged £1,000 of his own money, told ITV’s Good Morning Britain: “I think we will get it within 48 hours.”
He added: “It seems to me and many of my colleagues in the House of Commons patently daft that we have got the most iconic clock in the world – literally, it’s a world heritage site – that that should stay silent on this occasion.”
He also claimed, without providing any supporting evidence, that the £500,000 cost of reinstating the bell, which has been removed for restoration, had been “massively exaggerated” by parliamentary authorities because “officials in the House of Commons just don’t want to do it”.
On Tuesday, Mr Johnson said his government was working up a plan to allow people to “bung a bob for a Big Ben bong”, although his spokesman later said there were no plans for a government fund.
Big Ben has been silenced since 2017 because of renovations to the clock and tower.
Brexiteers’ calls for church bells to also ring out across the country to mark Brexit Day have received a similarly lukewarm reaction.
The body that represents bellringers, the Central Council of Church Bell Ringers, said bells should not be rung “for political reasons”, while the Bishop of Buckingham, the Right Rev Alan Wilson, said it would be “deeply divisive”.
He said: “Churches are there for the whole community, not for a political faction to crow over people they have beaten.”
Meanwhile Nigel Farage confirmed that the Leave Means Leave group has been given permission to hold an event in Parliament Square on 31 January.
The Brexit Party leader has also called for the government to fund the cost of Big Ben ringing on Brexit Day.
He wrote on Twitter: “The government have rejected my idea for them to underwrite the cost of Big Ben bonging on Jan 31st.
“If they were enthusiastic about celebrating Brexit and not embarrassed by it, they would make this commitment.”
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