In a statement Mrs Foster, whose party props up the British government at Westminster, said the European Union plan would effectively mean imposing a trade barrier between the Northern Ireland and the rest of the UK.
"I'm sure they too will be persuaded the alternatives - of no deal or potentially a Corbyn government - would not be of benefit to them or Northern Ireland", he said.
The issue of the Irish border, the major sticking point, was "close to being settled", it said.
Following three days of talks with key figures in Brussels, DUP leader Arlene Foster said Mrs May could not in "good conscience" accept the proposals now on the table from the European Union.
Numerous problems are now being solved on a step by step basis "but there are, of course" several big issues which we really need to get to grips with, ' the prime minister said.
On the other hand, May is expecting support from her inner Cabinet, including two prominent Brexiters: environment secretary Michael Gove and Brexit secretary Dominic Raab, the FT said.
Commenting on the Irish backstop, Barnier said the EU's plan to keep Northern Ireland in the Single Market and customs union would help keep the border invisible - a goal of both sets of negotiators.
The unionist leader said she feared that the European Union was forcing Northern Ireland into a deal that amounted to "the worst of one world" in which the region would remain part of the single market and remain under the ECJ, and would remain subject to trade restrictions with the rest of the UK.
Prized post-Brexit trade agreements with the United States, China and India would have only a "relatively modest" impact on Britain's economy, according to the government's official forecaster.
"Indeed, Northern Ireland's access to any new United Kingdom trade deals would also be regulated by Brussels", she said.
The DUP has threatened to vote against the government's budget if it goes ahead with the plan.
Theresa May has been put on notice by her allies in the Democratic Unionist Party to change course on Brexit or risk the collapse of her government.
Former Conservative leader and Brexit supporter Iain Duncan Smith warned the prime minister she should "listen very carefully" to the DUP, which he said echoed many Tory concerns.