Both parties expressed interest in forming the next government.
The Brexit-supporting party has boosted its number to 10 with two gains, giving it a potentially pivotal role.
May will need the support of another party or group of parties to command a majority.
Mr Corbyn, whose party performed better than many expected to win 262 seats in the election, told the Sunday Mirror: "I can still be prime minister".
The DUP strongly opposes marriage equality, leaving open the possibility of whether Davidson and other more socially liberal Tories might be alienated by the Conservative party's coalition partner. It does not care about domestic United Kingdom politics, save to the extent that it would prefer a strong and stable government that will not be overruled by its parliament, or overly influenced by other stakeholders who do not like the compromises that are required to be made. The European Union is still no wiser as to what the UK's negotiating position will be in the upcoming Brexit talks. "Now let's get to work", she said.
Senior DUP leaders said they moved quickly to cement a deal with May to prevent Corbyn from becoming prime minister.
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.
The President of the European Commission says the UK's election result is one of "particular importance". She didn't specify how her Conservative Party would "work together" with the Northern-Ireland-based party.
"The British have spoken, they have voted, and have given the Conservative party a majority, albeit a simple majority, which is something of a surprise", Prime Minister Edouard Philippe said on France's Europe 1 radio station. Theresa May did not have to call this election. "That's not a matter for me", she said.
With 649 of 650 seats declared, the Conservatives had won 318 seats and Labour 261.
"There is no doubt that the Conservative Party failed to match the initial expectations of a convincing landslide", Matthew Goodwin, professor of politics and global relations at the University of Kent, England, told ABC News via email one day before the election. It would make British exports more price competitive around the world - something that was reflected in a rise in shares for major British companies on Friday, with the FTSE 100 index was up 0.6 percent.
Ahead of her coalition announcement, main opposition Labour Party leader Jeremy Corbyn said May's attempt to win a bigger mandate had backfired and called on her to resign.
But in practice, a working majority will require just 322 MPs, as the Speaker does not vote and Sinn Fein has so far declined to take up its seats.
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