Both sides have vowed they will not impose a "hard" border between EU-member Ireland and British-ruled Northern Ireland, but almost a year since formal exit proceedings began, no agreement has been reached on how to manage Britain's only land frontier with the EU.
The EU draft says Britain will "contribute to and participate" in the bloc's budgets for 2019 and 2020, which covers the transition period.
EU Chief Negotiator Michel Barnier claims it's needed.
Ireland's Indian-origin Prime Minister Leo Varadkar hit back at hard Brexiteers like UK Foreign Secretary Boris Johnson for failing to come up with any workable solution to the border issue.
Mrs May argued that the proposal, set out by the EU's chief negotiator Michel Barnier, would risk breaking up the United Kingdom by creating an internal trade barrier.
"I am anxious by the time, which is short", he told reporters, referring to an October target for agreeing a treaty, including a transition period, in time for it to be ratified before Brexit in March 2019.More news: Luka Modric Charged with False Testimony in Former Dinamo Zagreb Exec's Trial
"The draft legal text the commission have published would, if implemented, undermine the UK common market and threaten the constitutional integrity of the UK by creating a customs and regulatory border down the Irish Sea, and no UK prime minister could ever agree to it", she said in one of the most fiery Brexit clashes to date between London and Brussels. "It is for all of us to work together".
It will detail three key topics, including the future of the Irish border and procedures for a fallback option that Northern Ireland would remain in "full alignment" with the EU's rules and regulations, if no better solution is found.
Major, who campaigned to stay in the European Union, said in a report yesterday that it must be parliament, not the Government, that makes the final decision on any new deal with the European Union, and that there is an argument for a second referendum on any such deal.
Responding, Labour anti-Brexit MP David Lammy tweeted "God help us all this isn't just stupidity and ignorance but wilful recklessness", while Paul Blomfield said it was "unbelievable".
Meanwhile, former British prime minister John Major made a significant intervention with a speech on Thursday, calling for a second referendum to allow the public to change its mind.
Mrs May's stance on the Irish border was welcomed by the DUP leader, Arlene Foster, who denounced the Brussels scheme as "constitutionally unacceptable" and potentially "catastrophic" for Northern Ireland. European Union officials acknowledge that Britain has said there may be other ways to avoid border friction - but has yet to offer details.
But earlier on Thursday, May s spokesman asserted her aim of negotiating "a comprehensive and new economic partnership with the European Union where we can trade tariff-free and on as frictionless a basis as possible".