The current round of negotiations will continue until Thursday, Barnier said.
British and European Union envoys on Monday began a first round of negotiations on Britain's divorce from the EU with both sides saying it was high time to tackle details, though feuding within the London cabinet over Brexit terms may trouble the process.
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.
"We made a good start last month, and this week we'll be getting into the real substance", Davis is expected to say. "We need to examine and compare our respective positions in order to make good progress".
Earlier in March, British Prime Minister Theresa May had set a two-year timetable for leaving the EU.
The Brexit Secretary said progress had to be made this month on a deal for reciprocal rights for citizens.
Last week, the United Kingdom acknowledged for the first time it will have to pay a contribution upon leaving. The EU remains uncertain about whether British negotiators will be able to conclude an agreement, amid deep political divides in the government on Brexit. Chancellor of the Exchequer Philip Hammond exposed tensions within the British cabinet at the weekend by stating that transitional arrangements at the end of talks are likely to last a couple of years, far longer than the couple of months suggested by Trade Secretary Liam Fox. 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.
The British government issued what it called a "fair and serious" offer to grant EU nationals "settled status" but the offer was described by the European Council President as below expectations.
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.
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.