Theresa May also forced her two closest aides to quit on Saturday after they have taken the blame for the election campaign.
Senior Conservatives said there was no longer support in Parliament for a so-called "hard Brexit" after the party saw its Commons majority wiped out.
Some Conservative lawmakers called for May's joint chiefs of staff, Fiona Hill and Nick Timothy, to be sacked for their key role in the election campaign.
British Prime Minister Theresa May has reached an "outline agreement" with the ultra-conservative Democratic Unionist Party in order to be able to govern after a humiliating election that has left her authority in tatters.
Asked if No10 had made a mistake in briefing that a deal had been struck had strengthened DUP's hand, she said: "I'm not going to negotiate over the airwaves".
The resignations of Timothy and Hill, on whom May had been heavily reliant since her previous job at the interior ministry, will be a personal blow.
He also said he accepted responsibility for his part in the campaign and believed the party failed to speak to the voters who chose to vote for the opposition.
After confirming on Friday that her top five ministers, including Chancellor Philip Hammond, would keep their jobs, May must name the rest of her team, who will take on one of the most demanding jobs in recent British political history.
May said Barwell would help her "reflect on the election and why it did not deliver the result I hoped for".
Labour party leader Jeremy Corbyn is a good campaigner
The Conservatives finished the election with 318 seats, eight short of the 326 needed for a majority, and the party is now dependent on the 10 DUP MPs to ensure legislation can be passed through a hostile House of Commons.
Such an agreement would see the Northern Irish party's 10 MPs support the Conservatives on crucial votes when Parliament returns next week.
But the Protestant DUP was founded to defend Northern Ireland's place in Britain against demands by Catholic republicans for a united Ireland. But the ballot-box humiliation has wounded her leadership just as Britain is about to begin complex exit talks with the European Union.
Its eurosceptic wing has always been a thorn in the side of Conservative prime ministers.
European Council President Donald Tusk has warned there is "no time to lose" in starting Brexit talks, after May on March 29 started the two-year countdown to ending Britain's four-decade membership. The party has also criticised May for DUP deal. She later confirmed she had received assurances from the Conservative leadership that there would be no roll-back of LGBT rights as a result of the DUP alliance.
One cabinet source said May had offered them a reassurance that things "will be changing", and a remark in her television interview that she would announce further personnel changes was read by insiders as a hint that she could be ready to sacrifice Hill and Timothy. The referendum passed, and Cameron resigned the next morning.
May sought to frame the campaign around her personal leadership heading into Brexit, but this was undermined by public performances derided as robotic. This is more than the total of Labour, Lib Dems, SNP, Plaid Cymru and Green Party. She's taking us back to those times.
Meanwhile, Labour leader Jeremy Corbyn, fresh from his party's strong showing, continues to put pressure on the weakened Conservatives.
CGTN's Dan Williams reports on a dramatic day in London.
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