Crucially, last month, Mr Davis caved in to the EU's insistence that the talks would move on to trade only when "enough progress" had been made on Brussels' three priorities.
Barnier said: "We will now delve into the heart of the matter".
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".More news: UK, Australian leaders visit site of Borough Market attack
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.
Mr Davis is facing strong opposition from the EU over Britain's proposals for rights of European nationals living in the United Kingdom, as well as pressure to accept paying a huge "divorce" bill.
Earlier last week, British Prime Minister Theresa May's government published a draft law to formally end Britain's membership of the European Union, but opposition parties and the leaders of Scotland and Wales threatened to block what they called a "naked power-grab".
Fox also said it would be "foolish" to go into the Brexit negotiations without being prepared to walk away, saying Britain's negotiating partners needed to believe Britain would do so rather than accept a bad deal.More news: Pakistan army chief 'analysing' evidence against Kulbhushan Jadhav: ISPR
Brexit hardliners in the Cabinet and on the Tory backbenches are furious with Mr Hammond for championing a two-year transition deal to cushion the impact of leaving the EU.
Theresa May last month published a "fair and serious" offer to guarantee the future rights of the 3.2 million European Union citizens living in the United Kingdom and the 1.2 million British ex-pats in the EU.
All of the topics have created controversy, as European Parliament lawmakers criticized the U.K.'s "settled status" offer on EU residents in the United Kingdom, threatening to veto any Brexit deal if the offer isn't improved.
The British side had urged over the past months an immediate start of trade talks, but Barnier had insisted that key issues of Brexit must be dealt with before trade talks begin.More news: Bakayoko complete Chelsea deal
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