The emails, between senior Government officials, were passed to The Observer newspaper.
In one of the emails, a senior advisor says: “Arlene Foster said the DUP was ready for the possibility of a no-deal scenario, which she now believed was the likeliest one.
“She described Barnier as being difficult and hostile in her meeting today.”
The advisor who wrote the email, which was sent to senior Government figures, added he didn’t know whether Ms Foster was delivering an honest warning or exaggerating to increase pressure on negotiators.
The emails are particularly important because the Government currently relies on the support of 10 DUP MPs for its Commons majority.
Earlier this week, the DUP threatened to vote against the upcoming budget if Theresa May agrees to trade barriers between Northern Ireland and the rest of the UK, in a move which could bring down the Government.
Talks between the UK and EU remain at loggerheads over whether the backstop period, when the UK will remain aligned with the European single market, should be time-limited.
The backstop will come into place if the UK and EU can’t agree to a comprehensive deal on their future trade relationship.
It is intended to stop any hard border being needed between Ireland and Northern Ireland.
The plan has been condemned by some Brexiteers, who argue it would leave the UK subservient to EU rules and unable to sign its own trade deals.
In a piece for the Belfast News Letter, former Foreign Secretary Boris Johnson, argued: “If we let this go it will be the greatest national humiliation since Suez.
“The only way to put things back on the right track is to ditch the backstop and then to chuck Chequers.”
Several Cabinet Ministers are demanding any UK alignment with the customs union is clearly time-limited, to prevent the UK indefinitely abiding by EU rules.
However, Brussels is refusing to approve a time-limit.
A number of ministers including Esther McVey are believed to be “seriously considering their positions” on this over the weekend and could resign in protest.
Even if a deal is agreed the Government may struggle to get it through Parliament.
Analysis by the Edelman public affairs consultancy concluded that Mrs May can only rely on the support of 277 of her 314 Conservative MPs, and may well need some Labour support to get her Brexit deal through Parliament.