She remains under pressure from Tory MPs to seek concessions from the EU.
Michael Gove - who last week turned down the job of Brexit Secretary following the resignation of Dominic Raab - Andrea Leadsom, Penny Mordaunt, Liam Fox and Chris Grayling were all reported to be unhappy with the terms over the Northern Ireland border.
On Monday evening, the DUP voted in favour of a Labour amendment to the Budget, and abstained on other votes throughout the week.
With just over four months before Britain's departure, May was meeting European Commission President Jean-Claude Juncker to try to finalise an outline of future relations before a summit of EU leaders on Sunday to endorse the Brexit accord.
Not everyone who had pledged to send in letters has done so, the source said.
Even if they do, it's quite possible the United Kingdom itself will not.
It emphasised that Gibraltar (as well as the other overseas territories and the Crown dependencies) would not be excluded from negotiations on the future relationship and said they would get a deal "that works for the whole United Kingdom family".
"There isn't a majority in the House of Commons to allow that to take place". "If Westminster agrees why would we be against it", said Borrell.
On Monday, the DUP displayed their power by failing to back May's minority government in several votes on a finance bill.
The DUP has repeatedly said it would not accept any extra Northern Ireland-only checks.
But the fabric of that arrangement is certainly torn... and once faith is broken between the two, it's hard to see how it could be restored.
Writing in the Daily Telegraph, Lord Hague said: "I have lived through more crises in British politics than I can remember, but I have never witnessed one more serious than this". The cozying up of Dodds to Corbyn is an ominous sign for May, relying as she does on the DUP's support to pass legislation.
But there remain challenges to overcome before then.
He told the Irish parliament: "It's not the best of both worlds that the DUP wants".
May defended her Brexit deal in an article for Tuesday's Belfast Telegraph, saying it "puts Northern Ireland in a fantastic position for the future".
"Until it is clear. we will not be able to give our agreement", he warned.
Mrs May has insisted the agreement reached is in the national interest.
Opposition to the agreement is also building in the pro-Brexit camp.
The Taoiseach added that the draft agreement reached on the withdrawal of the United Kingdom from the EU should not be seen by unionists as a threat to the union.
"This is what needs [to] be made clear, and until it is clarified in the withdrawal agreement and in the political declaration on the future relationship, we can not give our backing [to the deal]".