Downing Street said an outline agreement on a "confidence and supply" arrangement had been reached and will be put to the Cabinet for discussion on Monday, reported Independent.
May had been hoping to boost her mandate for Brexit negotiations which are due to begin in just 10 days' time, but hung parliament has left Tories weakened with less seats that fell below the 326 needed to form a majority government.
However, the Democratic Unionist leader and first minister of Northern Ireland, Arlene Foster, said she wants to "bring stability to our nation" by backing Theresa May and the Conservatives to continue in power the Guardian reported.
Foster has informed that she was entering discussions with May over the details of any arrangement that would prop up a minority government.
Foster said the election in Northern Ireland, which saw 10 DUP MPs, including two new ones, elected to the Commons, was a "great result" for the union.
Theresa May has said sorry to the Tory MPs and ministers who lost their seats due to her decision of holding elections.
DUP and May's party share cordial relations since she became prime minister 11 months ago and are likely to enter into an agreement so as to stop Jeremy Corbyn becoming prime minister.
"We want there to be a government. We have worked well with May. The alternative is intolerable. For as long as Corbyn leads Labour, we will ensure there's a Tory PM," the Guardian quoted a DUP source.
The Tories won 318 seats, down 12, and will have to rely on the DUP to get things done. If some Conservative MPs desert the party on key votes than May's plans would be left in tatters.
Jeremy Corbyn, the Labour leader, has urged May to resign as he said she should "go and make way for a government that is truly representative of this country".