John Redwood, a senior Conservative, disputes the claim, arguing the result did not mean there should be any softening in a hard divorce from Europe, noting that Labour, the second largest party, also endorsed Brexit in its election manifesto.
He said his party would seek to vote down May's Queen's Speech, or program for government, when she presented it to parliament on June 19, and another national election might be needed to break the deadlock.
Survation, the opinion polling firm that came closest to predicting correctly the election's outcome, said a new poll it conducted for the Mail on Sunday newspaper showed support for Labour now 6 percentage points ahead of the Conservatives. Numerous key cabinet posts have already been declared as unchanged from the previous government, including Philip Hammond as Chancellor of the Exchequer, Amber Rudd as home secretary, Boris Johnson as foreign secretary, David Davis as Brexit secretary and Michael Fallon as the in-charge of the ministry of defence.
"I look forward to meeting with him again tomorrow if his appointment is reaffirmed, but yes I think it's an important issue that you raise - the objectivity of both governments, and both governments working strictly in accordance with our legal responsibilities under the Belfast Agreement, the Irish government as co-guarantor, indeed the British Government as co-guarantor".
He told the Sunday Mirror: "I can still be prime minister".
The DUP itself later issued a statement saying the talks had been positive, but stopped short of confirming a deal had been sealed.
Earlier today, May's joint chiefs of staff Nick Timothy and Fiona Hill resigned amid intense Tory criticism in the wake of the snap election which backfired spectacularly.
The right-wing British press speculated Sunday that Foreign Secretary Boris Johnson, a former journalist and the former Mayor of London, was plotting a leadership coup.
May announced the party would try to work with Northern Ireland's Democratic Unionist Party, an alliance that is fraught with difficulties.
Downing Street initially said an outline agreement on a "confidence and supply" arrangement had been reached with the DUP which will be put to the Cabinet for discussion on Monday. She confirmed this to German leader Angela Merkel in a phone call on Saturday. "At such a critical time, the Prime Minister must be clear with the people about the deal she has stitched up with the DUP behind closed doors". I think she will have to go unfortunately. "I'm sure many of them will want to vote for government measures needed to execute our departure from the European Union". Former Conservative cabinet minister Owen Paterson, asked about her future, said: "Let's see how it pans out".
However, there have been serious doubts on the stability of such a government and May's own leadership of the Tory party. Some say her failure means the government must now take a more flexible approach to the divorce.
The DUP is a socially conservative party that opposes abortion and gay marriage, and many of its members also are sceptical about man-made climate change and reject the theory of evolution.