"For us it's incredibly important that we now make good progress", Davis told reporters. They want to frustrate Brexit'.
May's decision to activate the EU's exit clause three months ago means that the United Kingdom will leave the bloc with or without a deal in March 2019.
Barnier, who has repeatedly called on weakened British Prime Minister Theresa May to quickly set out her divorce strategy, said they needed to "examine and compare our respective positions in order to make good progress".
Working groups will focus on three areas: citizens' rights; the EU demand that Britain pays some 60 billion euros ($69 billion) to cover ongoing EU budget commitments; and other loose ends, such as what happens to British goods in EU shops on Brexit Day.
The Brexit Secretary said progress had to be made this month on a deal for reciprocal rights for citizens.
Mr Hammond lifted the lid on cabinet feuding by claiming damaging stories about him had come from fellow ministers out to get him because he is pushing for a softer Brexit.
"If you want my opinion, some of the noise is generated by people who are not happy with the agenda that I have, over the last few weeks, tried to advance, of ensuring that we achieve a Brexit which is focused on protecting our economy, protecting our jobs and making sure that we can have continued rising living standards in the future". Chancellor of the Exchequer Philip Hammond exposed tensions within the British cabinet at the weekend by stating that transitional arrangements at the end of talks are likely to last a couple of years, far longer than the couple of months suggested by Trade Secretary Liam Fox.
Negotiation on the exit bill might be especially tough, following British Foreign Minister Boris Johnson's remark last week that the European Union could "go whistle" over its "extortionate" bill demand.
Discussions on the politically charged issue of the Irish border issue will be led by Sabine Weyand, Barnier's deputy, and Olly Robbins, permanent secretary of the department for exiting the EU.
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.
On Sunday, the former Tory chair Chris Patten warned that the Brexit deadlock represented one of the bleakest moments in British postwar history.