Barnier said after welcoming Davis that it was important to tackle uncertainties caused by Brexit.
It comes as Boris Johnson sought to calm high tensions by claiming a deal can be sealed with "profit and honour" on both sides.
Representatives for the United Kingdom (U.K.) and the European Union (EU) kicked off Brexit negotiations in Brussels on June 19, nearly 12 weeks after Britain's prime minister triggered the mechanism that would let it depart from the EU.
Only when "sufficient, concrete progress" on the first phase has been made will Mr Barnier recommend to the European Council that the negotiations can enter the next stage, taking in the future trading relationship, with that recommendation possibly coming at October's summit of EU leaders.
He said these are the rights of EU citizens living elsewhere in Europe and the problems of a EU-UK border, particularly in the context of Ireland. Article 50 stipulates the divorce deal should take account of the departing state's "future relationship with the union". Anything but the smallest "divorce payment" would be very hard politically for the government.
For his part, Michel Barnier, the EU's chief negotiator, promised a "constructive attitude" to the talks and saying that a "fair deal" was possible.
"The UK will remain a committed partner and ally of our friends across the continent", Davis said.
Mr Barnier had previously used hiking imagery to urge Prime Minister Theresa May - a fellow hiking enthusiast - to tread carefully on Brexit.
May herself will also have a chance to update the other 27 European Union leaders on her Brexit plans at a summit meeting in Brussels on Thursday.
It comes after Chancellor Philip Hammond made some signals towards a softer Brexit, hinting at a longer transition period to any new status after business groups demanded the United Kingdom stay in the customs union. "The U.K. has chose to leave the EU". May, whose future is uncertain after she lost her Conservative majority in an election this month, has insisted that trade talks start immediately and run in parallel. The other European Union countries have a united position, but the British are "in chaos", Weber added.
"The only possibility of a soft Brexit is if the European Union itself fundamentally changes, if the European Union allows member states to reclaim more control over their destiny, which is something I've advocated for many years". "I think it was recognition by all sides that the clock is ticking and we do really need to push on with this now and start to make positive strides towards getting a deal that is in both sides' interests".
"For me, it is above all about the EU27 proceeding together and listening carefully to Britain's wishes and expectations", Merkel said after meeting Romanian President Klaus Iohannis.