"We will now delve into the heart of the matter".
Barnier, who has repeatedly called on Britain to set out a full divorce strategy, said they needed to "examine and compare our respective positions in order to make good progress". Barnier and Davis are to brief the media on Thursday, when they would give political endorsement to whatever officials have managed to agree.
The British side had urged over the past months an immediate start of trade talks, but agreed last month that key issues of Brexit be dealt with first.
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. Other issues, such as the cost of Brexit, are also on the table.
Further talks are expected in late August and the autumn ahead of a Brussels summit in late October, where European Union leaders will decide whether the United Kingdom has made "sufficient progress" on the Brexit divorce to allow trade talks to go ahead.
The EU has demonstrated increasing confidence in recent weeks, accusing Britain of dithering over whether it wants a "hard" or "soft" Brexit more than a year after the shock referendum that propelled May to power.
"The European Parliament will reserve its right to reject any agreement that treats EU citizens less favourably than they are at present", the letter said.
Davis added that apart from citizens' rights and the UK's exit bill, Northern Ireland issues are also a priority of these talks.
He said negotiators had made a good start last month but that they would now be working "on the substance of the matter". The Sunday Times cited five unidentified sources as saying Hammond had described public-sector workers as "overpaid" during the cabinet meeting last week, and was criticized by Johnson for the statement.
It said Britain's plan "would cast a dark cloud of vagueness and uncertainty over millions of Europeans" by giving EU citizens living in Britain fewer rights than those of British citizens in the EU.
Finance minister Philip Hammond, who like May campaigned a year ago to keep Britain in the European Union, accused unnamed colleagues of trying to undermine what is seen as his push for a "soft Brexit" that prioritizes trade rather than hardliners' demands for controls on European Union immigration or an end to European Union legal oversight.
Hammond told the BBC that government ministers were becoming increasingly convinced of the need for transitional arrangements to reduce disruption, with Brexit looming.