Downing Street says it hopes to finalize the deal next week, after Parliament resumes sitting.
Earlier, May's former head of communications launched a series of brutal attacks on the prime minister's decision to place so much trust in her two chiefs of staff and her inability to connect with voters or the rest of the party.
In return for shoring up the Tory majority the party has demanded considerable more resources for Northern Ireland, more influence and involvement in trade deals. "The DUP are experts at playing this game".
"I don't think we should talk about some prolongation of the deadline", he said in Prague.
However, another hinted it could cause Mrs May problems in Westminster, as it may commit her to spending on public services in Northern Ireland that she cannot hand out elsewhere.
Such a move would be strongly opposed by the Irish nationalists of Sinn Fein who share power in Belfast under a 1998 peace agreement that ended three decades of violence between nationalists and pro-British loyalists.
"History will show, alliances between Ulster unionism and British unionism has always ended in tears", he said. "Do your best to avoid a "no deal" as a result of 'no negotiations, '" he advised.
DUP leader Arlene Foster confirmed that she had spoken to May and would be talking with the Conservatives about "how it may it be possible to bring stability to our nation at this time of great challenge".
"I don't think we should read these results as calling into question the stance on Brexit which was clearly expressed by the British people", he added.
"We need a government that is capable of action, which can negotiate Britain's exit... the British need to negotiate their exit but with a weak negotiating partner, there is a danger that the talks are bad for both parties", European Union budget commissioner Guenther Oettinger told German radio.
However, senior DUP figures made clear they were looking at a limited "confidence and supply" arrangement - rather than a more formal coalition - leading to some MPs to predict that there could another general election before the year is out.
Amid a febrile atmosphere among Tory MPs, the Foreign Secretary, Boris Johnson, is reported to be preparing a new bid to become Prime Minister as Mrs May's grip on No. 10 becomes increasingly fragile.
As expectations mounted that Labour could draw up its own Queen's Speech, the party leader vowed "we can still do this". The Conservatives won more than 13.6 million votes, which is an historically high number, and more than Tony Blair won in all three of his election victories.
The Prime Minister has promised the Scottish Conservative leader Ruth Davidson that any deal between their party and the Democratic Unionist Party will not damage gay rights in the UK. "I just don't see how she can continue in any long-term way".
But it would require the political agreement of all 27 other European Union member states and they would demand a high price.
"I sought, and to be fair to the prime minister, received a categoric assurance that in talking to the DUP that there would be no suggestion of any rollback on LGBTI rights in the rest of the United Kingdom", she said.
If Mrs May stands down as prime minister, any negotiations over Brexit may have to be delayed while the Conservative Party chooses a replacement for her and discusses if or how to change its approach to Brexit.
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