Big Ben's tower, in the Palace of Westminster, has been under renovation since 2017 and the work is not due to be completed until 2021.
A public fundraiser could allow Big Ben to chime for Brexit after MPs decided the £500,000 cost could not be justified.
But an attempt to have it enshrined in law that Big Ben would chime for Brexit failed last week.
Boris Johnson yesterday said he is planning the crowdfunding campaign after Parliamentary authorities balked at the £500,000 cost.
Still, his latest nationwide plan could use a little bit of translating before Americans - or perhaps anyone - can understand it. Johnson is launching a new £500,000 push so the United Kingdom can "bong Big Ben on Brexit night", and he wants Britons to "bung a bob for a Big Ben bong" so that can happen.More news: Russian prime minister submits resignation to President Vladimir Putin
He told The Telegraph: "Having seen the Prime Minister's interview on BBC Breakfast, I am delighted to learn that the Government will be establishing a crowdfunding scheme so that people around the country who wish to contribute to marking this historic act of freedom can do so voluntarily". The cost includes restoring the clapper, installing a temporary floor in the belfry, testing, ringing, and the knock-on costs of delaying conservation works.
But Mr Francois did not share that view, saying to the paper: 'What was the cost of it chiming on New Year's Eve?
"Did this cost £500,000 on each occasion? Let the bells ring out for Brexit!"
The Prime Minister's official spokesman said: "If the public wants Big Ben to bong and the money is raised then that is great".
Pro-Brexit campaigners are hoping to mark Britain's departure.More news: Priyanka Chopra and Richard Madden team up for Russo Brothers' 'Citadel'
Sir Lindsay Hoyle said the great bell in Parliament's Elizabeth Tower should be rung on Jan 31 if MPs "wish to do that", stating that the UK's formal departure from the European Union on Jan 31 will be a "significant moment".
To mark Brexit day, the Leave.EU group wants "patriots to ring the bell of their local church.to celebrate Britain's new-found independence!"
Fellow party member Neil Gray added: 'The obsession with Big Ben chiming for Brexit really highlights the shallowness of the Vote Leave campaign.
"There are historical moments for which bells have been rung - end of world wars for example. However, the central council, as a principle, does not endorse bellringing for political reasons", they said.
Interim leader of the Liberal Democrats - which campaigned during the December 2019 election to cancel Brexit altogether - Ed Davey has complained that the Brexiteer should not be allowed to organise the party, claiming it would not be "unifying".More news: Smoke From Australia’s Fires Will Make ‘full circuit’ Around the World