Barnier says the two sides must reach agreement on a deal by October 2018, to give the European and British parliaments time to ratify the deal by Brexit Day in March 2019.
DUP leader Arlene Foster insisted that any deal with the Conservatives would not undermine the province's peace accords.
"We will build the broadest possible consensus for our Brexit plans and that means giving Parliament the maximum amount of time to scrutinise these bills".
Brexit Secretary David Davis starts negotiations in Brussels later on Monday that will set the terms on which Britain leaves the European Union and determine its relationship with the continent for generations to come.
The British Pound was stable against the US Dollar through Monday's Asian session as markets await the long-delayed start of talks with the European Union to frame the terms of "Brexit".
The newspaper quoted another former minister as saying: "If she weakened on Brexit, the world would fall in. all hell would break loose".
The negotiations have been billed as the most complex in Britain's history as it unravels 44 years of membership and its threat to walk out with no deal in place has anxious European capitals.
The European Commission's chief negotiator Michel Barnier has insisted initial talks will focus on the status of EU nationals in the United Kingdom and Britons in the EU, as well as the "divorce bill" of upwards of £60 billion and the issue of the Northern Ireland border with the Republic.
They will hold more than seven hours of official talks.
Issuing joint comments as they head into the first meeting, Davis said the talks were taking place in a "positive and constructive" tone, and that "there is more that unites us than divides us".
"If we're going to radically change the way we work together, we need to get there via a slope, not via a cliff edge", he said.
British Prime Minister Theresa May has said that the vote one year ago was partly a vote for Britain to control its borders, and has said that Britain will leave Europe's single market, as membership is incompatible with restricting immigration.
In a letter to Business Secretary Greg Clark, they also ask the government to "put the economy first".
Once sufficient progress has been made on the withdrawal issues, the European Union say they will open discussions on a new trade agreement.
A poll on Sunday also found majority support for a softer form of Brexit.
May reportedly faces "civil war" inside her Cabinet, over whether she should continue with her hardline position on Brexit.
Scottish Conservative leader Ruth Davidson had been among leading figures within the Conservatives and opposition parties arguing that the government's loss of its majority should force a rethink on the approach to Brexit.
But with news emerging that there are 7,000 outstanding issues ministers need to resolve prior to the UK's March 2019 exit date, it is apparent politicians failed to comprehend the enormity of quitting the European Union when they voted for Britain to have a referendum on the issue.
In a scathing attack on May's failed election campaign, Hammond also told the Andrew Marr Show that he had been prevented from selling the government's record on the economy. There is more that unites us than divides us. I think they are ready with it, but whether they are politically ready to do it I don't know.