"For us, it's incredibly important we now make good progress", added Mr Davis, who held a first day of talks a month ago to agree an agenda, a year after Britons vote to leave the EU.
Citizens' rights, Northern Ireland, the Brexit bill, and separation issues will be discussed this week.
Mr Barnier said: "We will now delve into the heart of the matter".
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. That was a reminder of a gulf in perceptions across the Channel where European Union leaders have assumed from the outset that Britain would need more than the two years allowed by treaty to negotiate the deal it wants to retain close, open trading links with the continent.
Hammond, a potential rival to May, acknowledged that ministers were divided on many elements of Brexit, after weekend newspapers were filled with reports of infighting.
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. "We need to examine and compare our respective positions in order to make good progress", Barnier told reporters.
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. Asked point blank on Monday if the cabinet was "split on Brexit", Johnson simply said he was pleased negotiations had begun and then defended the offer May has made to protect the rights of European Union citizens in Britain.
"We're seeing the division at cabinet level laid increasingly bare", said Ned Rumpeltin, head of currency strategy at The Toronto Dominion Bank in London, who sees the pound falling to US$1.26 (RM5.40) by the end of this quarter.
The rowing will be seen as further evidence of Theresa May's weakness after seeing her Commons majority wiped out in last month's general election.
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. 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.