Mr Barnier and British Brexit Secretary David Davis will open the talks on Monday at 11 am in Brussels with the aim of wrapping up a deal by the end of 2018 so that it can be ratified by the European and British parliaments before the United Kingdom leaves the bloc in March 2019.
DUP leader Arlene Foster insisted that any deal with the Conservatives would not undermine the province's peace accords. Foster said talks with May would continue into next week.
"We will go into speak with Sinn Féin again on Monday morning because devolution works and works for everybody in Northern Ireland", she said emphatically.
On Brexit, Ms Foster said her party wanted to see "a sensible Brexit and one that works for everybody". The party has refused to give a time frame for reaching a deal, though May is due in Brussels for an European Union summit on June 22-23 when she will want to show she has a solid grip on power.
Prime Minister May wants to negotiate the divorce and the future trading relationship with the European Union before Britain leaves in March 2019, followed by what she calls a phased implementation process to give business time to prepare for the impact of Brexit. The talks are taking place three months after Prime Minister Theresa May formally triggered the two-year Brexit timetable.
Chancellor Philip Hammond said the Government's priority in the negotiations should be to protect jobs, economic growth and prosperity.
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She also said that the talks with May's party had covered corporation tax and same sex marriage.
May's plans included Britain's departure from the single market, as well as spelling out that no deal would be better than a bad deal, as far as the government was concerned.
"We believe that the withdrawal process can not be concluded without the future relationship also being taken into account".
Philip Hammond, Britain's finance minister, says protecting the British economy should be the main goal of upcoming negotiations over the country's exit from the European Union.
"Michel Barnier, the European Commission's Chief Negotiator, and David Davis, Secretary of State for Exiting the European Union, agreed today to launch Article 50 negotiations on Monday, June 19", read the joint statement issued by the European Commission and Britain's Department for Exiting the European Union.
But after a generation of discord over Europe inside her party, May's future could depend on her ability to please both the eurosceptic and pro-European factions in her party.