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 European Union land border separates the British province from European Union member Ireland to the south.
And his allies blasted Michael Gove for reportedly being behind the "smear campaign" of leaking details from last week's Cabinet meeting.
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.
Liberal Democrats leader Tim Farron seized on the splits, saying: "The Conservative government is more interested in tearing itself apart than getting on with the monumental challenge it has set itself on Brexit".
He defended his position on Sunday and hit back at his critics. 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.
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.
Hammond told BBC1's Andrew Marr show: "If you want my opinion, some of the noise is generated by people who are not happy with the agenda that I have".
He said negotiators had made a good start last month but that they would now be working on the substance of the matter.
"I think on many fronts it would be helpful if my colleagues - all of us - focused on the job in hand".
The prime minister had made similar pronouncements before, but the statement to MPs and peers carried weight in Brussels.
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. Unsurprisingly, the Brits have expressed opposition to these figures; Foreign Minister Boris Johnson labelled the sums "extortionate", saying Brussels can "go whistle" for the money. These two key players will meet regularly through the week, with the aim of keeping the talks on track.
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.
According to EU negotiating papers, the European Commission, leading the Brexit talks for the EU, wants the ECJ to be able to hear a case if "facts relating" to it occurred at any point before Brexit day.
On Sunday, the former Tory chair Chris Patten warned that the Brexit deadlock represented one of the bleakest moments in British postwar history.