Downing Street aides ascertained that Johnson is planning to bring about post-Brexit changes to his team before the Commons rises for recess on 13 February.
The Prime Minister's proposal came after the StandUp4Brexit's "Big Ben must bong for Brexit" campaign started this week.
Mr Francois said it is "inconceivable" that the clock used to mark the UK's departure from the European Union "could be any other than the most iconic timepiece on Earth, which is Big Ben".
While the House of Commons Commission (which oversees the fix work) told the AP that making sure Big Ben could ring to signify the moment the United Kingdom leaves the European Union, it "could result in huge costs to the public purse". It is hard to say when.
"If we want to get rid of it ... we'll replace it with the Trump deal. It doesn't take two weeks to put a temporary floor in and attach a clapper to a bell".
Mr Johnson appears to have rowed back after initially backing the calls for the bell to ring with a Downing Street spokesperson saying the Prime Minister is focusing on official plans to mark the occasion.
An attempt by some MPs to enshrine these chimes into law was dismissed in the U.K.'s Parliament last week.
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In a video posted on the fundraising site, Mr Francois said he is "delighted to announce that our Eurosceptic colleagues from StandUp4Brexit have agreed to host a crowdfunding campaign to help raise half a million pounds so that Big Ben can chime at 11pm on the 31st of January".
But in an increasingly heated and farcical atmosphere, it subsequently transpired that the government had no such plan, and the House of Commons commission had no way of accepting public donations.
"Any novel form of funding would need to be consistent with principles of propriety and proper oversight of public expenditure", it said.
Asked whether people should contribute to the appeal, the Prime Minister's official spokesman said: "I'm just reflecting that the Parliamentary authorities have set out some potential problems".
The reported move is believed to be part of a plan that adviser Dominic Cummings approved to promote hard work in the government and a fast track to make good on Whitehall's Brexit promises.
The new Speaker of the Commons, Lindsay Hoyle, who chairs the commission, said, "you're looking at £50,000 a bong".
Brexiteers were motivated to dip into their pockets after Mr Johnson said he was "working up a plan so people can bung a bob for a Big Ben bong" as the bell is now silenced while renovations are carried out.