Reports say the Democratic Unionist Party of Northern Ireland is prepared to support May without a formal coalition agreement.
May's left-wing Labour rival Jeremy Corbyn, once written off by his opponents as a no-hoper, said May should step down and he wanted to form a minority government.
Across the United Kingdom, out of 649 seats declared at the time of writing, the Conservative party won 318, a loss of 12 from the 2015 election. The precise nature of the relationship they are targeting is not yet clear.
British Prime Minister Theresa May and her husband Philip stand on the doorstep of 10 Downing Street, London, after addressing the press Friday, June 9, 2017 following an audience with Britain's Queen Elizabeth II at Buckingham Palace where she asked to form a government.
In the coming weeks, in Foster's words, the DUP "will do what is best for Northern Ireland.' Many political commentators on the mainland are asking, somewhat nervously, just what price that might entail: roads, schools, hospitals?"
The party is now more secular and attracts a wider demographic than when it was founded, shifting from fundamentalist outsider to political pragmatists.
The party has often found itself embroiled in controversy over its stance on gay rights issues.
And although it has a female leader, it remains an overwhelmingly male, white bastion and is certainly more conservative than the Conservatives.
While climate change scepticism is not official party "policy", the DUP has previously appointed a denier as environment minister in N Ireland, and it counts a number of creationists among its senior members.
Beijing can help tame North Korea, says US
China and Japan both claim islands in the East China Sea. "In short, the rules-based regional order is under challenge", she said. China, Pyongyang's ally, backed the sanctions but renewed its appeal for dialogue to ease tensions on the Korean peninsula.
As it now seems clear the DUP will prop up the reeling Conservative government in Westminster, that accord demonstrates that while the party is rigid in its convictions it can also embrace a flexible pragmatism.
After Brexit, the border between Northern Ireland and the Republic of Ireland becomes an European Union frontier and the DUP is not in favour of a so-called hard border.
Going back decades, the DUP was at the vanguard of the failed Save Ulster from Sodomy movement that campaigned against the 1982 legalisation of homosexual sex in Northern Ireland.
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. Sinn Fein, with seven elected MPs, has similarly wiped out the Westminster representation of the moderate nationalist SDLP.
In the South Belfast constituency won by the DUP, voters expressed dismay at the obliteration of the center ground.
The pro-Brexit, pro-Unionist party had a good election, seeing its overall vote rise by 10% to 292,316 votes. Watch out for the margins for Nigel Dodds & Gavin Robinson if they win and can Emma Pengelly gain South Belfast, if she does the DUP could be heading for a record result equaling or besting their 2005 result.
Mr Paterson instead suggested the focus would be on Brexit.
The Northern Irish party backed Brexit in last year's referendum campaign and are fully committed to the decision to leave the European Union.
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