LONDON (REUTERS) - British Prime Minister Theresa May plans to put her programme to parliament next week, a sign of confidence that talks on propping up her government will conclude in a deal and end days of uncertainty since she lost her majority.

The prime minister has been holding talks with Northern Ireland's Democratic Unionist Party (DUP), keen to get the backing of their 10 lawmakers in parliament to help her pass laws and govern as Britain starts talks to leave the European Union.

Her failure to win a majority has put May under pressure over her Brexit plans from inside and outside her party and has prompted complaints about her choice of partner due to the DUP's stance on social issues such as gay marriage.

On Thursday, May visited the wreckage of an apartment building in west London, where at least 17 people were killed in a fire, and ordered a public inquiry.

The political fallout of the fire, which began late at night and trapped dozens of people in their beds, is also hurting May's Conservatives, who run the local council and have been blamed by the opposition for spending cuts that have starved budgets for building maintenance and safety checks.

A Conservative Party source said it was time for the government to get on with its business. "The talks are ongoing, they are very positive, they are constructive. There is a steady dialogue between the two sides that has never stopped at any point. It continues and when the deal is done, it will be done," a senior source in the Conservative Party said on Thursday. "We never put timescales on when we expect a deal to be done and I'm not going to start now."

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 start of parliament has been delayed since last Thursday's election, a gamble May took to strengthen her hand in talks to leave the European Union but which has left her scrambling for a deal with the eurosceptic DUP to keep her in power.

May does not necessarily need a firm deal from the DUP before opening parliament and might hope that she would receive the necessary backing anyway.

But Jeremy Corbyn, leader of Britain's main opposition, said his Labour Party would not support May's Queen's Speech in the lower house of parliament to try to force her out of power through a vote of no confidence.

The Conservative source said: "We're confident of getting an agreement, we're confident that the Queen's speech will be passed."

BREXIT TALKS

The 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.

Brexit minister David Davis plans to go to Brussels on Monday to start the talks, which will reshape not only Britain's role in the world, but also that of a bloc praised by many for ensuring peace after World War Two.

May's government has said its Brexit plans remain the same, and will be pressing for close economic ties but a clear break with the bloc to be able to control immigration and restore sovereignty over British laws.

Some opposition politicians say that stance is no longer acceptable, with some in her own party pressing May to soften her position and focus on saving jobs after Britain leaves. "Look at what the Tories (Conservatives) have managed to do to the UK in the space of just one year, firstly calling a divisive and reckless EU referendum .. then having lost that gamble pursuing a hard Brexit path," Scotland's first minister, Nicola Sturgeon, said.

"And if that wasn't enough, calling an unnecessary general election purely for the self interest of the Conservative Party and having mucked that campaign up they are now putting the country in hock to the DUP."