Negotiators will break into groups discussing four key areas of priorities before a planned news conference on Thursday.
Barnier, who secured Davis's consent last month to the EU's broad structure for talks, wants to hold the Englishman publicly to whatever else has been agreed during the week, EU officials say.
Working groups will focus on three areas: citizens' rights; the EU demand that Britain pays some 60 billion euros ($70 billion) to cover ongoing EU budget commitments; and other loose ends, such as what happens to British goods in EU shops on Brexit Day.
A fourth set of talks, run by Davis and Barnier's deputies Oliver Robbins and Sabine Weyand, will focus on curbing problems in Northern Ireland once a new EU land border separates the British province from EU member Ireland to the south. Some of that will have to wait for clarity on future trade relations.
One key early advance that EU officials hope for this week is for Britain to stop challenging the principle it will owe Brussels money — though how much will have to be argued over and cannot be calculated until Britain actually leaves.
Three more weeks of talks, interspersed with internal EU sessions to coordinate the views of the 27 other governments, are scheduled, from late August until early October. At that point, Barnier hopes to be able to show "significant progress" on the divorce priorities for EU leaders to give him a mandate to launch negotiations on a future free trade agreement.
Davis and May had pressed over the past months for trade talks to start immediately but accepted the EU's sequence for negotiations last month. However, Brussels accepts that details on the divorce terms will still be open when trade talks begin.
In a sign British ministers are coming round to the EU view that a trade deal can at best be sketched in outline over the next 20 months, two members of May's cabinet who were on opposing sides of the Brexit referendum debate both said they expected some transitional phase to start in 2019 to smooth the passage from full EU membership to a final free trade pact.