Sterling snapped a three-day advance ahead of the talks that are likely to focus on protecting citizens' rights, which has been a key sticking point so far.
Earlier in March, British Prime Minister Theresa May had set a two-year timetable for leaving the EU.
"For us, it's incredibly important that we now make good progress", added Davis.
'That we negotiate through this and identify the differences, so that we can deal with them, and identify the similarities so that we can move forward.
The current round of negotiations will continue until Thursday, Barnier said.
"We made a good start last month but. we are now getting into the substance of the matter", Davis told reporters as he was welcomed at the European Commission in Brussels by Barnier, a little more than a year after Britons voted by a narrow majority to leave the EU.
Citizens' rights, Northern Ireland, the Brexit bill, and separation issues will be discussed this week.
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One of the most contentious has been the future rights of 3.2 million European Union nationals living in the United Kingdom, and the 1.2m United Kingdom nationals living 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.
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. At that point, Barnier hopes to be able to show "significant progress" on the divorce priorities for European Union leaders to give him a mandate to launch negotiations on a future free trade agreement.
Mr Davis made clear his first priority would be resolving the issue of citizens" rights, saying he was determined to make "real progress'.
Yet defining the role of the ECJ will be one of the most sensitive issues of the Brexit divorce treaty, as Theresa May has vowed to take the United Kingdom out of the jurisdiction of the Luxembourg court.
The Chancellor, Phillip Hammond, accused his rivals in the Cabinet of trying to undermine his plans for a business-friendly withdrawal over the weekend.
European diplomats fear the United Kingdom is continuing to box itself into an uncompromising position and are expecting it to push back strongly on the court, as both sides map out their differences in detail during the negotiations.