Ahead of the election, Northern Ireland's largest party made clear its preference was for a Tory rather than Labour government.
May's weakened position in the party rules out big changes to the Cabinet lineup.
According to one of the close allies of Boris Johnson, at least five UK Cabinet ministers are pushing for him to topple Theresa May as Conservative party leader and Prime Minister, but the foreign secretary denied any plans of a coup.
May said Barwell would help her "reflect on the election and why it did not deliver the result I hoped for".
The EU had wanted exit talks to begin on June 19, but they could now be delayed if the inconclusive election leads to political turmoil.
The 10 DUP MPs would provide vital support to Mrs May after her majority was wiped out as the snap election gamble backfired, leaving her with 318 seats. The main opposition Labour Party surpassed expectations by winning 262. It will be more hard for a Conservative government to walk away from the European Union without a trade deal, as May has threatened to do.
Given May's diminished state, many in the European Union are wondering how long she will be leading the country and - by extension - the Brexit negotiations.
Ruth Davidson, the Conservative leader in Scotland, said she had asked May for assurances that there would be no attack on gay rights after a deal with the DUP.
There is even a chance May could step down as Prime Minister at some point in the future, if she is unable to govern successfully with a worse parliamentary hand than she held previously.
"She is now reliant on the DUP and absolute loyalty from the Conservative party - loyalty that was not given to her over the Budget".
Mr Timothy said: "I take responsibility for my part in this election campaign, which was the oversight of our policy programme".
Her Conservatives struck an outline deal with Northern Ireland's Democratic Unionist Party (DUP) for support on key legislation, a humiliating outcome for them after an election meant to make them a dominant force.
Damian Green Takes Over Government Digital Leadership
Interestingly, May has chosen Damian Green - a close and important ally - as her first secretary of state. Downing Street has also revealed that Liam Fox has retained his role as global trade secretary.
DUP leader Arlene Foster with MPs at the Stormont Hotel in Belfast.
Downing Street and the DUP issued separate statements overnight, emphasising that the deal between them had not yet been finalised.
The two parties are broadly politically aligned, but it remains to be seen what price the DUP will demand for its support.
The arrangement with the DUP will make governing easier, but it makes some Conservatives uneasy.
Numerous party members are also uncomfortable with the kind of deal that will have to be struck with the DUP, a socially conservative party that takes an opposing stance on issues such as abortion and same sex marriages.
The strength of any deal looks set to be tested when the Commons meets, with Jeremy Corbyn vowing to try to bring down the Government by defeating Mrs May in Parliament and insisting: "I can still be prime minister".
Irish Prime Minister Enda Kenny tweeted Sunday that he had spoken with May "and indicated my concern that nothing should happen to put (the Good Friday Agreement) at risk".
Northern Ireland is now at a political impasse, after a power-sharing agreement broke down in January over a scandal involving a green energy scheme.
"There's a possibility of voting the Queen's Speech down and we're going to push that all the way".
We could end up paying money to Brussels, remain subject to European Court rulings and Commission directives and be unable to take back control over immigration.
"It's just how long she's going to remain on death row", former Conservative finance minister George Osborne, who was sacked by May when she became prime minister previous year, told the BBC.
Conservative MP Sarah Wollaston said on Twitter: "I will always oppose the death penalty & would resign if others imposed it".
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