Mr Davis went into the talks announcing that he was in a "positive and constructive" frame of mind and that he was determined to build a "strong and special partnership" with the EU.
"The most right-wing and reactionary party in northern Ireland will be backed by the most right-wing and reactionary major party in Britain, directly the product of British imperialism's history of intervention and domination in Ireland", he commented, pointing out that the UDA and UVF paramilitaries had backed the DUP in this month's General Election. "We need to get on with the job", said Grayling, after accepting that the election had a "disappointing result".
"I think it's fair to say that Northern Ireland is perhaps the most vulnerable part of Europe to a bad Brexit deal should that happen", he said.
Speaking in Luxembourg Friday before a meeting of European Union finance ministers, Hammond said, according to Reuters, "My clear view, and I believe the view of the majority of people in Britain, is that we should prioritize protecting jobs, protecting economic growth, protecting prosperity as we enter those negotiations and take them forward".More news: What we know: Pedestrians struck near north London mosque
Mr Coveney said he would highlight the particular issues facing Northern Ireland - in regard to the peace process and cross-border movement - when he met the EU's chief negotiator Michel Barnier in Luxembourg on Tuesday.
The pound rose slightly against the dollar, reaching more than $1.28 at noon, but fell against the euro today as Brexit negotiations finally began in Brussels.
Mr Coveney said a deal was "do-able" but there were differences between the parties that needed to be bridged.
Both leaders expressed confidence that the Stormont institutions could be up and running again by the deadline of June 29, averting a return to direct rule from Westminster.
"All of the messaging I am getting is people are up for a deal", he said.More news: Change of Prime Minister 'not on the agenda' - Chris Grayling
"As a UK Government we remain absolutely steadfast in our commitment to the Belfast Agreement, its successor agreements".
Northern Ireland has been without a powersharing executive since March and without a first and deputy first minister since January, after Sinn Fein collapsed the administration amid faltering trust and relations with the DUP.
According to a statement, Mr Davis is expected to say that while there is a long road ahead, the destination was clear - "A deep and special partnership between the United Kingdom and the EU".
The new landscape is a belated outworking of the two governments" strategy of cultivating the DUP and Sinn Fein, rather than parties that were, at the time, considered "centre ground'.
Sinn Fein has gained seven constituencies in the June 8 elections, with the DUP coming first with 10.More news: Sena will support NDA's presidential candidate
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