Following the loss of her parliamentary majority in last week's polls, May is now desperately seeking the support of the 10 MPs from Northern Ireland's ultra-conservative Democratic Unionist Party.
He told the Press Association the British government could not act as a "neutral facilitator" in Northern Ireland, as the Good Friday Agreement envisages, if it was dependent on one of the Northern Irish parties for its majority in the House of Commons.
Many in Northern Ireland, especially the Loyalist community, see this DUP-Conservative arrangement as a good thing if the deal is signed today, as it may highlight the areas of Northern Ireland that need investment.
The nationalist Sinn Fein and SDLP and the cross-community Alliance have all made clear Northern Ireland Secretary James Brokenshire can not chair the ongoing process to restore power-sharing at Stormont due to the perceived conflict of interest.
Demanding a swift start to Brexit negotiations under Article 50 of the European Union treaties, the former Belgian prime minister compared Britain's position to the heroine of Lewis Carroll's Alice In Wonderland, who found herself in a room with many doors and no idea what lay behind them.
The government is still in talks with a small Northern Irish party to secure the support of its 10 members of parliament to pass legislation.
Leadsom said: "The government has agreed with Buckingham Palace that the state opening of parliament will take place on 21 June 2017".
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Before her visit, Sinn Fein's leader at Stormont Michelle O'Neill said: "I will be making it very clear that any deal between the Tories and the DUP can not be allowed to undermine the Good Friday and subsequent agreements".
The EU's chief Brexit negotiator Michel Barnier also warned on Tuesday that time was passing. That's why we're ready to start very quickly. Sinn Fein MP Michelle Gildernew: "This new arrangement is very unsettling and people are concerned and anxious about what it may mean".
May made clear the Brexit negotiations would begin next despite uncertainty.
She will have to manage conflicting demands from within her own party, including a proposal for business groups and lawmakers from all parties to agree a national position for Britain's most complex negotiations since World War Two.
Brexit Secretary David Davis said he still expects the talks to begin next week and Theresa May insisted on Tuesday the timetable remains on track. "We will be asking the prime minister to be open with politicians and also with the public", SDLP leader Colum Eastwood said in a statement.
"There is very little time left". And he asked: "Is the populist illusion to limit free movement of people more important than the prosperity and fortune of the British workers, the British industry, the British companies and the British economy?".