An appeal to raise £500,000 by the weekend has been launched to ensure Big Ben chimes when the UK leaves the EU at 23:00 on 31 January.
The famous bell has only rung a few times since renovations began in 2017.
The PM’s spokesman said there could be “potential difficulties” using money raised from public donations.
But the Brexit Party’s Richard Tice said it would be “pretty feeble if we can’t organise for a bell to chime at this historic moment”.
StandUp4Brexit, the organisation behind the crowdfunder, says if it does not reach the target, the money will be donated to the Help For Heroes charity.
Appealing for contributions, the group writes: “However you may have voted in the referendum, this unique moment is unlikely ever to be repeated and if you would like to see it marked by the chiming of the most iconic timepiece in the world, please donate now.”
So far, total donations raised remain in the tens of thousands of pounds – with the largest single contribution coming from the Conservative MP Mark Francois, who has given £1,000.
Business Secretary Andrea Leadsom has said she donated £10.
Why would Brexit bongs cost £500,000?
The House of Commons Commission’s estimate is made up of two separate costs – bringing back the bonging mechanism and installing a temporary floor, and the cost of delaying the conservation work, says BBC’s Reality Check.
On the former, the commission says the floor in the belfry has been removed – work that began on 2 January in order not to interfere with New Year’s Eve.
The reconstruction work on the floor is likely to be significant, involving resurfacing and waterproofing.
The Commons has ruled out getting this done by 31 January, which is why a temporary floor would need to be installed and then removed.
Read more here.
What have the politicians said?
On Tuesday, Commons Speaker Sir Lindsay Hoyle said: “The commission believes it is important to weigh up the costs this would involve if Big Ben is to chime on 31 January.
“You are talking about £50,000 a bong. We also have to bear in mind that the only people who will hear it will be those who live near or are visiting Westminster.”
But speaking to the BBC’s Today programme, Brexit Party chairman Richard Tice said he did not believe the figure was correct, pointing out that the bell rang on New Year’s Eve.
He also suggested that “bureaucrats in the Houses of Parliament” might stop the money being used on the grounds that it would not be public money.
He added that if the target was not reached, a recording of Big Ben’s chimes would be played through a loudspeaker at a Brexit Day celebration organised by Leave Means Leave group in London’s Parliament Square.
For Labour, shadow leader of the Commons Valerie Vaz said the fundraising effort was “an unprecedented approach”, adding that “any novel form of funding would need to be consistent with principles of propriety and proper oversight of public expenditure”.
Leader of the Commons Jacob Rees-Mogg responded: “One shouldn’t look a gift horse in the mouth.”
“If people wish to pay for things, that should be considered as part of their public spiritedness, rather than thinking everything should fall on the hard-pressed taxpayer,” he said.