It was a humiliating outcome after an election that May had planned to strengthen herself ahead of the Brexit push.
The two aides had been blamed for the Conservatives' disastrous result in Thursday's parliamentary elections. But Johnson said he backed May.
Conservative Party MP Anna Soubry said Ms May's position was untenable long term.
The change may help to quell some of the unrest within the party. "This is still on", Corbyn told the Sunday Mirror newspaper.
The Conservatives now plan to reach a so-called confidence and supply agreement with the DUP, which would involve it supporting a Conservative minority government on key votes in parliament but not forming a formal coalition.
That result has obliged a thrashed Tory party, now without an overall majority, to form a working relationship - effectively an alliance - with the Democratic Unionist Party or DUP. Although Arlene Foster said her party fundamentally rejected the endorsement of anyone involved in paramilitarism in response to the Loyalist Communities Council backing of the party.
If she is to succeed in delivering the end of Britain's European Union membership which 52 percent of the British public demanded a year ago, she must find a way to recapture the full support of her party because she will need their votes to pass legislation preparing for and ultimately enacting the departure.
The British pound tumbled on Friday against the US dollar and the euro before stabilising, down 1.7 and 1.4 percent against the two currencies respectively.
Before the election, she had been widely expected to sack finance minister Philip Hammond following a reported clash over her Brexit strategy.
British Prime Minister Theresa May to form a minority government
The Northern Ireland party is officially opposed to same-sex marriages and abortions, both which are legal in the rest of Britain. May's party is short of the 326 it needed for an outright majority and fairly down from the 330 seats it had before the election.
Others have also said a Conservative-DUP deal could endanger Northern Ireland's peace settlement, which relies on the British Government being a neutral arbiter between those who want the province to remain in the United Kingdom and those who want it to become part of the Republic of Ireland. "But we are nine days off from the Brexit talks starting", he told BBC Radio. She had a lead of 20 points, and the consensus view was that Labour leader Jeremy Corbyn was horribly unpopular and his party doomed.
The Survation poll for the Mail on Sunday finds that Labour would win 45% of the vote to the Conservatives' 39%, if voters were sent back to the polls. "It's just how long she's going to remain on death row", said former Conservative finance minister George Osborne, who was sacked by Ms May when she became prime minister previous year. Labour stunned even its own supporters by taking enough seats from the Conservatives to deny them a majority. "May fights to remain PM", said the front page of the Daily Telegraph, while the Times of London said: "May stares into the abyss".
"The Conservatives have not yet broken the British system of democracy, but through their hubris and incompetence they have managed to make a mockery of it", it said in an editorial.
A stream of senior lawmakers entered her office at 10 Downing St. on Sunday afternoon, to learn what roles they had been given in government.
Its eurosceptic wing has always been a thorn in the side of Conservative prime ministers.
On Brexit, the DUP supports leaving the European Union but opposes a return to a "hard" border with Ireland - which could happen if May carries through her threat to walk away from the talks rather than accept a "bad deal".
She has said she favours retaining the greatest possible level of access to Europe's single market. However, the DUP later said talks would continue.
Meanwhile sources close to the Foreign Secretary dismissed reports about his leadership ambitions as "complete bollocks" and suggested "someone is trying to make mischief".
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