Asked whether she could rely on DUP votes to pass her Chequers plan, a spokesman for the Prime Minister said: "The confidence and supply arrangement that we have with the DUP is a matter of record".
Barnier has amended the EU's "backstop" plan to try to minimise checks on goods coming into Nothern Ireland and make sure they take place away from the border.
DUP leader Arlene Foster insisted on Tuesday that her party would not accept customs or regulatory checks on goods travelling in either direction between Northern Ireland and Great Britain after meeting European Union chief negotiator Michel Barnier in Brussels.
For customs and VAT, Barnier said, "we propose using the existing customs procedures to avoid doing checks at the transit points".
However, the European Union says that means there must be some checks on goods coming into Northern Ireland in order to prevent those that do not meet European Union standards from going across in the Republic of Ireland and then reaching the continent.
And he told fellow MPs: "If we stay on our current trajectory, we will go into the next election with the Government having delivered none of the benefits of Brexit, with the country reduced to being a rule-taker from Brussels, and having failed to deliver on a number of promises in the manifesto and in the Lancaster House speech".
Sources in the party - whose 10 MPs are propping up the PM in power - have warned it would be "unacceptable" for Northern Ireland and mainland Britain to be "treated differently".
The EU negotiator insisted "checks" would be needed between the United Kingdom and the bloc in future, and they could not happen between Northern Ireland and the Republic.
"An agreement is within reach for October 17, next Wednesday, if we succeed to the end of this negotiation now", Barnier said in a speech to business leaders at the European Parliament.
A Brexit sign is seen between Donegal in the Republic of Ireland and Londonderry in Northern Ireland at the border village of Muff, Ireland, February 1, 2018.
Former Brexit minister Steve Baker said on Tuesday that as many as 40 hardcore Tory Brexiteers will defy the party whip should they be forced to vote in favour of the Chequers plan. But he stressed: "Brexit was not our choice".
With just six months before Britain leaves the European Union in its biggest shift in trade and foreign policy in more than 40 years, both sides say they are intensifying talks to try to avoid a "no deal" Brexit that could harm the world's fifth largest economy.
If 40 of her lawmakers voted against a possible deal, the fate of the government and exit process would depend on the opposition Labour Party, which has indicated it will vote against nearly any deal May might secure.