A partnership between the Conservatives and the DUP may threaten the ability of the British government to be a neutral broker between the unionists and the nationalists, Mr Jonathan Powell, former chief British government negotiator on Northern Ireland, wrote in The Guardian.
Sinn Fein has said the prospect of a deal between the Conservatives and the DUP is causing anxiety and fear, with the party warning it could have implications for power-sharing talks in Northern Ireland.
The DUP leader said: "There's been a lot of commentary around the issues that we are talking about and it won't surprise anyone that we are talking about matters that pertain, of course, to the nation generally". The pair left separately, with May going to parliament for the election of the House of Commons speaker.
May desperately needs the DUP's 10 seats to pass legislation.
Mrs Foster arrived with colleague Nigel Dodds and waved to reporters in Downing Street but refused to be drawn on whether she would agree to a deal on Tuesday.
May is due to travel to Paris later on Tuesday to meet with French President Emmanuel Macron but a representative for her Conservatives will continue the talks with the DUP, the source said.More news: Golden State Warriors beat Cleveland Cavaliers to win Championship
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A number of deadlines to reach an agreement have already fallen by the wayside since March's snap Assembly poll, which was triggered by the implosion of the last DUP/Sinn Fein-led administration over a dispute about a botched green energy scheme.
The talks with the DUP follow her apology to Conservative rank-and-file lawmakers in a meeting Monday which signaled she would be more open to consultation, particularly with business leaders demanding answers about the details on Britain's departure from the European Union.
The Prime Minister did not mention the ongoing deliberations as she addressed MPs but called on Parliament to "come together in a spirit of national unity" to deal with the challenges facing the country.
He joked that he welcomed the prospect of a Queen's Speech once this "coalition of chaos has been negotiated", but said if this did not happen, he was "ready to offer strong and stable leadership in the national interest".
Currently, the Tories and the DUP are considering a "confidence and supply" arrangement which would see the loyalist party back the Government to get its Budget through and on confidence motions.
Some MPs have expressed disquiet at the Conservatives tying their fortunes to a party which is opposed to equal marriage and has expressed its support for tightening the laws on abortion - but No 10 has insisted these issues will not be up for grabs in any way.