Ahead of the first detailed negotiating sessions, Barnier and his opposite number Brexit secretary David Davis said that the menu for the talks would include Citizens' rights, the U.K.'s financial obligations to the European Union - known as the Brexit bill - separation issues, and Northern Ireland.
"We need to examine and compare our respective positions in order to make good progress", Michel Barnier, the EU's chief negotiator said ahead of the meeting.
"For us it's incredibly important that we now make good progress", Davis told reporters.
Both men declined further comment, with Barnier saying they would brief the media on Thursday after their teams have spent four days tackling a range of priority issues.
Earlier on Sunday, finance minister Philip Hammond had said senior British government ministers were becoming convinced of the need for transitional arrangements to reduce disruption as Britain leaves the EU. At that point, Barnier hopes to be able to show "significant progress" on the divorce priorities for European Union leaders to give him a mandate to launch negotiations on a future free trade agreement.
This week's talks are also set to address more detailed concerns such as Britain's future in Euratom, the EU's nuclear safety agency, and the role of the European Court of Justice, the EU's top court.
Too many no-win scenarios with North Korea
This latest test-launch further demonstrates a sobering reality: the threat of North Korea is growing, and it's advancing quickly. Many North Koreans are unsure of the missile's true abilities, though, a source living in North Korea's Yanggang province said.
May's minority government remains fragile one month after the snap June 8 election in which her Conservative Party lost its majority, forcing it to seek an alliance with Northern Ireland's small ultra-conservative Democratic Unionist Party.
Mr Grayling said the suggestion that there were "profound and fundamental differences" between Cabinet ministers on Brexit were "a bit exaggerated", but admitted: "We're not a group of clones, we have discussions around the Cabinet table and outside Cabinet, we debate issues, we decide what's right and then we get on with it".
"I hope very much that people will look at the offer in the spirit it deserves".
Barnier's deputy, Sabine Weyand, and the permanent secretary of the United Kingdom department for exiting the EU, Olly Robbins, will focus on the issue of Ireland on Tuesday and meet regularly throughout the week.
Trade minister Liam Fox, who favours a cleaner break with the EU, said on Sunday he could live with a transition - during which it is likely Britain would keep paying a share of the EU budget and follow EU rules - as long as it was kept short. The pound fell from a 10-month high against the dollar on concern that discord within the United Kingdom government is worsening before the nation starts the second round of Brexit negotiations with the European Union.