"We are starting this negotiation in a positive and constructive tone, determined to build a strong and special partnership between ourselves and our European allies and friends for the future", he said.
Mr Davis, who unlike Ms May has long campaigned to leave the EU, will meet chief EU negotiator Michel Barnier, a former French minister, at the European Commission's Berlaymont headquarters at 11am (0900 GMT). The U.K. will exit the European Union in March 2019 unless all European Union nations agree to extend the negotiations.
The EU's chief negotiator, Michel Barnier, said he hoped they could agree a format and timetable on Monday.
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.
He listed the importance of reaching a rapid deal on the status of European Union nationals and the border between Northern Ireland and the Republic as priorities, but did not mention the size of the UK's "exit bill" for leaving the bloc, which is bound to be an early sticking-point.
While the European Union negotiating team led by Michel Barnier has been ready for months, Britain stalled even after it triggered the two-year process on March 29.
Australia beat New Zealand to qualify for CT semis
England play their last group match against Australia, which would be a do-or-die match for the Steve Smith-led side, on June 10. Williamson top scored for New Zealand with 87 from 98 balls, hitting eight fours.
Both sides were keen to stress they were entering the talks, which they hope will reach agreement in autumn 2018, in good faith.
But May's mandate for a hard Brexit has since eroded after her Conservative party lost its parliamentary majority in the June 8 general election. "The U.K. has chose to leave the EU". "We have N. Ireland new executive, New Irish Taoiseach and United Kingdom administration, I will be meeting new Irish foreign minister in Luxembourg", added Barnier. Before the Tories' election debacle, the Brexit secretary threatened to play hardball with Brussels, vowing to resist the EU's timetable of deferring trade talks until after progress was made on the divorce.
The aim is to make sufficient progress to get leaders of the other 27 European Union countries to agree to move on to talks on a future relationship with Britain, including a trade deal, he said.
Barnier however said that "a fair deal is possible and is far better than no deal - that is what I said to David today".
Austrian Foreign Minister Sebastian Kurz said: "If we don't succeed, both sides will lose".
A year after Britons shocked the continent by voting on June 23 to cut loose from their main export market, new debate within Prime Minister Theresa May's cabinet on precisely what kind of trading relationship to pursue has perplexed European Union leaders, who warn time is tight to agree terms before Britain leaves in 2019.
Scattered showers to return Monday afternoon
Given the clouds and periodic rain, it won't be as hot Monday , and highs should stay in the mid/upper 80s. Showers and thunderstorms are possible Friday with a frontal boundary crossing the region.
Still, Johnson called on people to look at the more distant future.
The EU appear to have claimed the first victory of the talks.
Barnier has warned that the negotiations must be wrapped up by October 2018 to allow time for all parties to ratify a final accord by March 2019.
"The new Northern Irish executive needs to be set up in a few days' time, at the same time we have a new government and a new Taoiseach in Dublin and of course there are the ongoing political discussion in London which we are also following closely".
Other official at the table included Glynn Williams - director general at the Home Office, Mark Bowman - director general, worldwide finance at HM Treasury, Simon Case - director general, UK-EU partnership team, Alex Ellis - director general at the Department for Exiting the European Union, Jane Walker - aide to David Davis and Christian Jones - press secretary to David Davis.
India captain Virat Kohli: Pakistan completely outplayed us in Champions Trophy final
Hardik Pandya came to bat at five down and his 76 runs from 43 balls nearly brought the Indian team back into the game. England's Ben Stokes, of course, was born in New Zealand . "It has been an up and down ride".