The Japanese car manufacturer, which is the UK's fifth biggest, is expected to make the announcement on Tuesday.
It is thought that the plant will close in 2022, but it is thought to be keeping its European headquarters in Bracknell.
The factory, which is Honda's only one in the EU, produces 160,000 Civic vehicles a year, more than 90 per cent of which are exported to Europe and the US.
Local Tory MP Justin Tomlinson said he and fellow Tory MP Robert Buckland had spoken to the Business Secretary and the car firm about the decision.
He tweeted: "They are clear this is based on global trends and not Brexit as all European market production will consolidate in Japan in 2021.
"Working with Honda, Gov (led by the Business Secretary), staff and Unions there will be a taskforce set up to provide support for all staff (as we did when jobs were lost previously at Honda).
"Honda will be consulting with all staff and there is not expected to be any job losses, or changes in production until 2021."
Green MEP Molly Scott Cato tweeted about the "shocking news" for her constituents.
She said: "Shocking news. My thoughts are with the many people I represent in Swindon who work for Honda. We’ll be finding out more details as soon as we can.
"We’ve heard Honda’s concerns about Brexit ever since the referendum but this has come totally out of the blue."
Lord Alan Sugar added: "Honda to leave UK with a potential loss of 3500 jobs in 2022. It won't be long for Nissan to make same decision.
If true, this is simply devastating news. Brexit is destroying British manufacturing before our very eyes. Parliament must secure an opportunity for the British people to call a halt to this madness. https://t.co/oBPp1rFd0k— Owen Smith (@OwenSmith_MP)February 18, 2019
"Sunderland you tipped the scales in 2016 vote and now you will pay with mass unemployment in your area not just Nissan factory but all the peripheral suppliers."
In a statement last month, Honda announced a six-day post-Brexit shutdown as part of its preparations for disruption.
They added: "This is to facilitate production recovery activity following any delays at borders on parts.
"These contingency provisions have been put in place to best mitigate the risk of disruption to production operations at the Swindon factory."
Announcement of #Honda Swindon factory closure is a shocking blow to the Government's delusional belief that Brexit won't destroy British industries & won't cause thousands of job losses
Ministers need to wake up to the truth now, there is no Brexit dividend only #Brexitdisaster— Tom Brake MP (@thomasbrake)February 18, 2019
Last year, senior vice-president of Honda Europe Ian Howells told the BBC that a no-deal Brexit would cost the company tens of millions of pounds.
Ian Howells told the BBC that quitting the EU without an agreement would affect the carmaker's competitiveness in Europe.
In September, he said the Japanese firm was preparing for no-deal, but had not discussed relocating the Swindon plant.
Honda were not immediately available to comment, while the Government declined to comment.
Honda set to close Swindon car plant - the car manufacturers are going. A pity that the Brexit Lie manufacturers don’t all leave and let us keep honest jobs. And yes Swindon area where Honda are based voted for Brexit. Time to vote again. Very sad. https://t.co/7neYTmNe9Y— Gavin Esler (@gavinesler)February 18, 2019
Unite national officer for the automotive sector Des Quinn said: “If these reports are confirmed, this would be a shattering body blow at the heart of UK manufacturing.
“The car industry in the UK over the last two decades has been the jewel in the crown for the manufacturing sector – and now it has been brought low by the chaotic Brexit uncertainty created by the rigid approach adopted by prime minister Theresa May.
“We are seeking urgent clarification from Honda on the implications of these serious reports.
“The 3,500-strong workforce do skilled, well paid jobs that the UK can ill-afford to lose.
“We will be doing everything we can in the coming days and weeks to support our members at this grave time for them, their families and the UK economy. This will also affect thousands of jobs in the extensive supply chain across the country. This would be the single biggest automotive closure since Rover in 2005.
“Business secretary Greg Clark needs to make an urgent statement on what the government intends to do to rectify this dire situation, if the reports prove correct. It appears that the chickens are coming home to roost big-time because of Brexit uncertainty.
“If the government had advance warning of this dreadful news and did not alert the unions, this is an appalling and cavalier attitude by ministers.”