Though on the surface, Thursday's meeting with Northern Irish parties is aimed at breaking the logjam in forming a new cross-party regional government in the province, May needs broader acceptance of a Conservative-DUP deal.
Ceasefire negotiations began in the 1980s, eventually culminating in the Good Friday Agreement, which was negotiated by the governments of the UK, US and Ireland, as well as representatives of Northern Ireland groups.
Through the 1970s-1990s, the DUP opposed successive accords that proposed Dublin's involvement in power-sharing arrangements, including the Good Friday Agreement, which ended decades of conflict and has maintained peace across Ireland.
The party brought the name of the late Martin McGuinness into the talks yesterday by handing over a letter.
Mr Eastwood made a number of proposals to Theresa May aimed at restoring power sharing government in Northern Ireland.
Labour is believed to be considering tabling amendments to Wednesday's speech on a range of issues including school cuts, preserving the winter fuel allowance, and scrapping the bedroom tax, which some Conservatives may allow to pass by abstaining on the vote.
The Conservative source said the talks to leave the European Union would not be delayed, removing the question mark over the negotiations being derailed by May's lack of a parliamentary majority lost in an election she did not need to call.
However, British sources insist she is "confident" of getting her will and delivering the policy programme she wants.
The DUP have also rejected the idea of a "special status" for Northern Ireland arising out of Brexit, even though that is precisely what is needed.
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Mr Killen said he would use his speech to the Pride event in the capital to celebrate progress on LGBTI rights law, while warning Mrs May not to roll back on it to please the DUP.
Both the government and the DUP have refused to implement key agreements on language and equality rights and dealing with the legacy of the past.
Conservative Party sources say May wants to show her government is up and running but her loss of authority in last week's election will make it harder to handle a hectic agenda - Brexit talks with the European Union, tackling a slowing economy, a political crisis in Ireland, and a devastating fire in London.
It comes amid concerns the Government will compromise its stated impartiality in the region if it enters a confidence and supply deal with Mrs Foster's party at Westminster.
The EU has said it wants the rights of EU citizens in the United Kingdom to be the "first priority" of Brexit negotiations.
"The risk is that Northern Ireland continues to fail to find solutions, and potentially the peace process unwinds", he told AFP.
"We also respect the other parties' mandates, we want to get back to an executive that has all the parties around the table to collectively take decisions".