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.
While Mr Barnier will hold a press conference at the close of talks on Thursday, Mr Davis' team has yet to confirm whether the minister will join him.
Mr Barnier said: "We will now delve into the heart of the matter".
Barnier has dismissed the British rights offer as falling short of the European Union demand that its 3 million citizens there keep all their existing rights for life and have recourse to the European Union courts to enforce those rights even after Britain has left.
Protecting the rights of all our citizens is the priority for me going into this round and I'm clear that it's something we must make real progress on. The EU remains uncertain about whether British negotiators will be able to conclude an agreement, amid deep political divides in the government on Brexit.
Mr Davis returns to Brussels this morning for round two.
The talks are the first full round of negotiations that formally began last month with a one-day session to agree on a timetable.
The Chancellor, Phillip Hammond, accused his rivals in the Cabinet of trying to undermine his plans for a business-friendly withdrawal over the weekend. Her spokesman said she would tell ministers not to reveal cabinet discussions.
Hammond himself acknowledged that ministers were divided on other elements of Brexit.
"I do think on many fronts it would be helpful if my colleagues - all of us - focused on the job at hand", he told the BBC.
The negotiations this week are expected to focus on the post-Brexit rights of citizens in each other's nations, the bill Britain has to pay to meet existing commitments, the border issue in Ireland and the pre-eminence of the EU's Court of Justice.
In London, media were rife with reports of infighting echoing the Leave-Remain rifts that May's Conservative party suffered during the European Union referendum.
Foreign Secretary Boris Johnson then fanned the flames when he said in the Commons that Brussels could "go whistle" if it expected the United Kingdom to pay a hefty "divorce bill" in respect of its outstanding financial obligations. Negotiators aim to guarantee, for example, that a consignment of vehicle parts ordered before Brexit Day could still be subject to European Union product rules - and the ECJ - even if it reached its intended destination after Britain has left the EU.
United Kingdom finance minister Philip Hammond said Sunday that Britain will take responsibility for the money it owes, but dismissed the 100 billion euro figure as "ridiculous".