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 GMT on 31 January.
The famous bell has only rung a few times since renovations began in 2017.
The PM's spokesman highlighted "potential difficulties" in 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 veterans' charity Help For Heroes.
More than £155,000 had been raised by Friday morning, with one of the largest single contributions coming from Tory MP Mark Francois, who donated £1,000.
Business Secretary Andrea Leadsom has said she donated £10.
Appealing for contributions, the group writes: "However you may have voted in the referendum, this unique moment is unlikely ever to be repeated...
"If you would like to see it marked by the chiming of the most iconic timepiece in the world, please donate now."
The House of Commons Commission has estimated that getting the bell to ring during renovation works on the Palace of Westminster's Elizabeth Tower, which houses it, would cost between £350,000 and £500,000.
The body - a group of MPs and officials responsible for the day-to-day running of Parliament - said this was because of the costs of bringing back the chiming mechanism and installing a temporary floor, and delays to the conservation work.
On Wednesday the commission said the estimated costs could not be justified, and appeared to cast doubt on the idea that public donations should cover them.
In a statement, the body said using donations for the works would be an "unprecedented approach".
"Any novel form of funding would need to be consistent with principles of propriety and proper oversight of public expenditure," it added.
On Thursday, the PM's spokesman said: "The House of Commons authorities have set out that there may be potential difficulties in accepting money from public donations.
"I think the PM's focus is on the events which he and the government are planning to mark 31 January. It's a significant moment in our history and we want to ensure that's properly recorded."
On Tuesday, Commons Speaker Sir Lindsay Hoyle said there was a need to "weigh up the costs" involved in making Big Ben chime for Brexit.
"You are talking about £50,000 a bong," he added.
"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 Brexit Party chairman Richard Tice told the BBC's Today programme 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 event organised by Leave Means Leave group in London's Parliament Square.
The event gained "provisional authorisation" from the Office of the London Mayor on Wednesday.
Responding to the call for donations, Commons leader Jacob Rees-Mogg told the Commons: "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."