Cabinet Office Minister David Lidington is also due to hold Brexit meetings with figures from the UK's devolved administrations on Thursday.
The British government and leading politicians in Northern Ireland insist they will not agree to a withdrawal treaty that foresees the possibility of Northern Ireland being kept inside the EU's economic area while the British mainland is not.
May gathered selected members of her cabinet late Thursday to update them on the Brexit talks, and the Financial Times reported that she told them that a deal was drawing closer.
The de-facto Deputy Prime Minister was pressed on the DUP's support when he appeared on ITV's Peston on Wednesday night.
She added that Northern Ireland was not being offered the "best of both worlds". "Many of her cabinet colleagues have assured me of their unionism", she said.
He reiterated the EU's line welcoming May's proposals for a bold free-trade deal "without tariffs or quotas" and close security ties after Brexit.
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In a move seen by some as a warning shot, DUP MPs failed to back the Government in voting against a Labour amendment to an Agriculture Bill outlining post-Brexit reforms on Wednesday night.
While talks with Brussels have intensified since May emerged from her infighting Conservative party conference last week, there has been no breakthrough.
Not such a pleasant morning for the Prime Minister who is trying to please the DUP, the European Research Group - the ultra pro-Brexit wing of her own party - and the EU.
Instead she proposed Britain as a whole stay temporarily aligned with the EU's customs union until a wider trade deal is agreed.
Earlier this week EU officials expected that a possible new United Kingdom proposal on the Irish border issue would suggest having the United Kingdom remain in the customs union indefinitely.
His constituency contains the port of Larne, identified by European Union chief Brexit negotiator Michel Barnier as a place where checks on animals and animal products from Great Britain into Northern Ireland could be intensified in order to prevent a hard border with the Republic of Ireland.
Eurosceptic Conservative MPs who want a clean break with the European Union have expressed alarm at reports that, as it races to get a deal, the government may agree this alignment would last indefinitely.