Theresa May at Downing Street with Nigel Dodds, Arlene Foster and Sir Jeffrey Donaldson of the DUP

The prime minister, who relies on a parliamentary deal with the DUP to prop up her minority government, stressed that she did not agree with the Northern Irish party’s opposition to gay marriage.

She insisted the Conservatives – which she once said was viewed as the “nasty party” – had not gone backwards by signing a pact with the DUP and pointed out the deal was not a formal coalition.

In an interview with LBC Radio, Mrs May – the daughter of a vicar – said the Church of England “will want to reflect” on whether to allow the clergy to bless gay marriages.

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Challenged on whether the deal with the DUP had taken the Tory party back decades, she said: “The Conservative Party hasn’t gone back but I can completely understand why people who are concerned about some of the attitudes that the DUP... some of the policies that the DUP promote, why they are concerned about this.”

She added: “I’m very clear that the Conservative Party in government doesn’t agree with the DUP on those issues, on issues on LGBT rights.

“We will continue to push forwards, to enhance LGBT rights and we are pleased with what we have been able to do so far and want to do more.

“That won’t be changed by our relationship with the DUP, which of course isn’t a formal coalition.”

But she defended the DUP deal, saying: “If you think back to what happened after the election, we were the only party who could form a government and there’s a responsibility to the country to ensure that it has a government, and a government that can put legislation through.”

Mrs May was asked whether her father, who died in 1981, would have been sympathetic to the idea of the Church of England blessing gay marriages.

She said: “I think as an individual he very much valued the importance of relationships, of people affirming those relationships and of seeing stability in relationships and people able to be together with people that they love.”

Pushed on whether there should be a change in the law, she said: “This has to be a matter for the Church ... the Church of England has itself come a distance in terms of looking at these issues and obviously they will want to reflect, as attitudes more generally change, as society changes.”

At a Downing Street LGBT reception, Mrs May said she believed same-sex marriage should be introduced in Northern Ireland.

“I think that LGBT people in Northern Ireland should have the same rights as people across the rest of the UK,” she said.

The PM also indicated she is preparing to reform laws to allow trans people to legally choose their gender without undergoing invasive medical checks.