Colum Eastwood, leader of the SDLP, Northern Ireland's second-largest Irish nationalist party, said his party had a "positive" meeting with the prime minister over her efforts to come up with a deal with the DUP.
Earlier, Andrea Leadsom, leader of the lower house of parliament, said the government had agreed with Queen Elizabeth, who reads out the new government programme, that the "state opening of parliament will take place on 21 June 2017".
The Irish republican party will join the other four main parties in Northern Ireland in Downing Street on Thursday, June 15 as Conservative talks with the DUP continue.
The EU's chief Brexit negotiator, Michael Barnier and the UK's Brexit minister, David Davis, will meet in Brussels on Monday (19 June), kicking off negotiations likely to take nearly two years. He pulled out of the event after a fire gutted a high-rise residential tower in London, leaving at least 17 dead.
"If the Government cannot even secure a deal with the DUP, how on earth can they get a deal with the EU?"
The proposed "supply and confidence" deal would see the DUP back the Conservatives in votes on the Budget and confidence motions.
The fact Davis agreed to an agenda on Monday that will see him and British officials spend some seven hours in the European Union executive's headquarters discussing who will talk to whom when and about what has come as a relief to some European Union negotiators."The big clash that we feared isn't there", said one senior official involved.
Northern Ireland's power-sharing administration, created under a 1998 Good Friday Agreement peace deal, collapsed in January and talks have been going on since a regional election in March to restore it.
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Previous Northern Ireland executives have wanted to bring it down to 12.5% to match the corresponding rate in the Irish Republic.
The concession sets the tone for negotiations over Britain's withdrawal from the European Union that are set to begin in Brussels on Monday.
Former prime minister David Cameron, who called last year's European Union referendum and resigned after losing it, told a conference in Poland that "there will be pressure for a softer Brexit".
A serious harm caused to the relations might be reflected in other spheres and affect the work of North Atlantic Treaty Organisation, which is in no party's interest, he said. May would remain the prime minister of the country while the DUP MPs would play a critical role in helping the Conservatives remain in power.
London-based banking giant HSBC said Friday that it would keep more jobs in Britain depending on the government's approach to Brexit.
Former prime minister David Cameron said May needed to listen to opposition parties, and that there would be pressure for a softer Brexit that would give greater priority to a close trading deal with the EU.
"In reality, the government is here simply because the DUP allow it to be so", she said.
Brussels has insisted talks on the so-called divorce, taking in issues including the fee the United Kingdom will have to pay to sever its ties, must make sufficient progress before any discussion on a future trade agreement could begin.