Mr Davis said there was a "long road ahead" but predicted a "deep and special partnership".
"When we eventually get to the stage where the council decide we have made enough progress, both sets of dialogue will continue, including free trade".
Chancellor and Tory MP Philip Hammond, meanwhile, dismissed Teresa May's controversial comment that "no deal was better than a bad deal" appearing on BBC's Andrew Marr show this weekend.
A Department for Exiting the European Union spokesman said: "We have been crystal clear about our approach to these negotiations".
In Luxembourg last week for a meeting of the 28 European Union finance ministers, Hammond underlined that he wanted both the European Union and Britain to take a "pragmatic" approach to the talks.
Anxious by immigration and loss of sovereignty, Britain a year ago voted to end its decades-old membership of the 28-country bloc - the first country ever to do so - in a shock referendum result.
Barnier said the two sides will have one week of negotiation every month, and use the time in between to work on proposals and exchange them.
European Union negotiator Michel Barnier said the negotiations which should lead to a breakup by March 2019 "must first tackle the uncertainties caused by Brexit - first for citizens, but also for the beneficiaries of the European Union policies and for the impact on borders, in particular Ireland".
Mrs May stressed the importance of Brexit negotiations as she was quizzed about ongoing pressure over her leadership following a disastrous election and criticism of her response to the Grenfell Tower disaster.
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Britain's negotiations with the European Union over its exit from the bloc begin on Monday and stand to be complicated by the surprise loss of Prime Minister Theresa May's parliamentary majority in a national election last week.
The UK Government has announced that due to the amount of legislative work involved in the Brexit preparations, the next parliamentary session will be extended from 1 year to 2 years.
The EU's chief negotiator, France's Michel Barnier, welcomed his counterpart David Davis with a handshake and smiles for the press in the European Commission's landmark headquarters in central Brussels flanked by the EU and British flags.
Having lost a majority in parliament, May is in talks with Northern Ireland's Democratic Unionist Party to secure the support of its 10 lawmakers to win any kind of vote, including on the pieces of legislation needed to enact Britain's divorce from the EU.
Echoing Barnier, British Brexit minister David Davis applauded the "very productive discussions", saying, "I've been encouraged by the constructive approach that both sides have taken".
She told BBC One's Sunday Politics: "Where you have politicians right across the European Union and the United Kingdom who share the desire for a successful outcome, with low tariffs, zero non-tariff barriers, free trade between ourselves, co-operation on security and so on, it should be perfectly possible to meet the time frame".
"Theresa is leading the government and I think the government needs to get on with the job", he said.
Monday's talks however are likely to focus on the practical details of timings for the coming months, with the big, divisive issues left aside for now, officials said.
If the United Kingdom opts for a Brexit in which it leaves the Single Market and the Customs Union, the Irish border would become a closed border unless negotiators can agree on creative solutions.
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