The Northern Ireland-based party is being courted by May to create an alliance to push through the Conservative Party's agenda after a disastrous snap election left May short of a majority in Parliament.
Theresa May's failure to win a majority at the General Election has led to suggestions her Brexit plan to leave the EU's Single Market and Customs Union could be altered, while rival parties have demanded involvement in exit talks following the inconclusive result.
However, putting the pro-British unionist DUP in a position of influence in London could also undermine the British government's ability, enshrined in a 1998 peace agreement, to function as an impartial broker between Northern Ireland's unionists and its Catholic Irish nationalists.
Following more than an hour of talks between May and DUP leader Arlene Foster on Tuesday, May said the discussion had been productive and Foster said she hoped a deal could be reached "sooner rather than later". But pressure was mounting on Mrs May to change course on the type of Brexit Britain should pursue, and consult more widely across rival parties.
A day after France and Germany both said Britain could reverse Brexit if it wanted, Verhofstadt said Britain's budget rebates and opt outs from European Union rules would disappear if it did so.
"Now a third of our parliamentary party is female and we have our most diverse group of MPs ever, I feel our MPs form a more representative group to elect a deputy leader".
The EU will keep the door open for Britain to return, but only on worse terms than it now has, European Parliament Brexit negotiator Guy Verhofstadt said Wednesday.
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"Yesterday Emmanuel Macron, the new French president, spoke about an open door, he said if Britain is changing its mind it will find an open door", Verhofstadt told the European Parliament.
"It continues and when the deal is done, it will be done", a senior source in the Conservative Party said on Thursday.
It follows warnings - including from former prime minister Sir John Major - that the Government will compromise its stated impartiality in the province if it enters a confidence and supply deal with the DUP at Westminster.
She added that Brexit, counter-terrorism and "doing what's right for Northern Ireland in respect of economic matters" were among the issues being discussed.
He joked that he welcomed the prospect of a Queen's Speech once this "coalition of chaos has been negotiated", but said if this did not happen, he was "ready to offer strong and stable leadership in the national interest".
A senior Brussels official said the amount, which compares to London's annual net European Union payment of around 10 billion euros, would still be "peanuts" in terms of the overall economy and also that the final bill would be determined less by technical and legal arguments than by hard-headed political horse-trading.
As European leaders tried to fathom exactly how Britain would begin the negotiations, German Finance Minister Wolfgang Schaeuble said Germany wanted a Brexit deal that would limit negative consequences for the bloc but also did not want it to weaken Britain.