Mrs May is seeking to rely on the Democratic Unionist Party's 10 MPs to win key votes in Parliament.
May however has vowed to stay on, and on Sunday unveiled a largely unchanged new cabinet, which met for the first time on Monday.
It opposes same-sex marriage and is anti-abortion - with abortion remaining illegal in Northern Ireland, except in specific medical cases.
After a lacklustre campaign from the Conservative party, and a surge in support for Labour leader Jeremy Corbyn, neither party achieved a majority and the country now waits in a state of anticipation and uncertainty. "I think the only thing that political commentators can agree on is that we have uncertainty right now and nobody has any clue what shape this negotiation is going to take".
The Conservatives and the DUP, "having enjoyed a strong relationship over many years", will work together, May said.
It was a surprise move - Gove was sacked as justice minister by May a year ago after his bid to become party leader forced now-Foreign Minister Boris Johnson from the race, amid accusations of treachery and political backstabbing.
Mr Osborne, who was sacked by Mrs May and now is editor of the London Evening Standard, said there was now no majority in the Commons for a "hard Brexit".
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.
May had planned a clean break from the EU, involving withdrawal from Europe's single market and customs union but some Conservatives and opponents hope the election shock will lead to a "softer" Brexit, which prioritises close trade links over controlling immigration. First by Brexit and now through this election.
It is likely the DUP will press for increased investment in Northern Ireland as the price of their support in Westminster and push for a more significant role in the Brexit process.
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The demonstrators appeared predominantly young - those who were born or grew up during Mr Putin's 17 years in power. We saw dozens of people plucked from the crowd - many of them young - and dragged roughly towards police buses.
Theresa May formed a minority government the same day the results were declared after meeting the Queen to stake claim.
Theresa May is clinging to power and facing a backlash from Conservative MPs over her election gamble that backfired.
There are also concerns about the potential impact of the proposed arrangement on Northern Ireland's peace agreement, which relies in part on London being an impartial arbiter between those, such as the DUP, who want the province to remain in the United Kingdom and those who want it to be part of Ireland. Conservative MPs are publicly airing their anger, some calling for her ouster and others demanding radical change in her style of leadership.
Concessions on such issues would seriously damage efforts by the DUP to secure a deal with Sinn Fein to restore Northern Ireland's devolved government, which collapsed in January.
British Prime Minister Theresa May is making appointments to her Cabinet as she tries to shore up authority undermined by a poor election result. May said that he would bring "considerable experience" to the post.
May has taken the blame for the Conservatives' relatively poor showing in last Thursday's election, in which the party surprisingly lost its majority.
Speaking at a meeting of the Conservative Party's 1922 committee, the Prime Minister said she would carry on in the job, saying: "I got us into this mess so I'm going to get us out of it".
"No one wants another election".
"What I'm doing now is actually getting on with the immediate job".
May acknowledged some of the mistakes she had made during the campaign, in particular the decision to announce the social care reform plans that were quickly dubbed the "dementia tax".
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