Update 1.30pm: Scottish Secretary David Mundell has said he wants the Democratic Unionist Party (DUP) to "change its position" on LGBTI issues.
British Prime Minister Theresa May is due to meet DUP leader Arlene Foster on Tuesday to finalise a deal on propping up her minority government.
Mr Mundell, Conservative MP for Dumfriesshire, Clydesdale and Tweeddale, was reappointed Scottish Secretary in a post-election Cabinet reshuffle on Sunday.
He became the first openly-gay Conservative Cabinet secretary when he came out in January 2015.
Questioned on the DUP's stance on gay rights, he said he does "not subscribe" to the party's position.
The North is the only part of the British Isles where same-sex marriage remains outlawed.
The DUP has repeatedly used a controversial Stormont voting mechanism - the petition of concern - to prevent the legalisation of same-sex marriage despite a majority of MLAs supporting the move at the last vote.
Mr Mundell told BBC Radio Scotland's Good Morning Scotland programme: "I don't subscribe to the DUP's position on these issues but the DUP will not be influencing these decisions within the rest of the United Kingdom.
"We're not in any way signing up to the DUP manifesto. Most of these issues are devolved to the Northern Ireland Assembly."
He added: "I would like to see the DUP change its position, and indeed Northern Ireland as a whole change its position, on LGBTI issues.
"Ruth Davidson has been very clear on that, she actually went out to Northern Ireland and set that out, so they can't be in any doubt where they stand on these issues.
"I think change is brought about, certainly in Northern Ireland, by persuasion, by people working together and the best way actually to achieve these is to get the Northern Ireland Assembly back up and running, and I hope that will also be possible."
First Minister Nicola Sturgeon said careful scrutiny of any deal with the DUP would be needed to guard against any rollback of equalities legislation and also raised concerns on the impact on the peace process.
Writing in the Daily Record, she said: "The Good Friday Agreement requires the UK Government to be an impartial broker between parties in Northern Ireland and it would be shameful if, in the Tories' pursuit of power, they jeopardised the chances of a return to devolved government in Northern Ireland."
Scottish Liberal Democrat leader Willie Rennie called on the Prime Minister to "abandon her embryonic alliance with the DUP before it does real and lasting damage".
He said: "Taking sides in the precarious politics of Northern Ireland could have significant ramifications for the whole of the UK including Scotland. The UK Government should be operating as an honest broker in Northern Ireland but that is impossible if it is in hock to one of the protagonists."
He added: "There is rightly anxiety about the DUP's views on abortion and gay rights and these views must not have an impact on government policy. Yet it is the constitutional threat that could have even wider and more immediate consequences."
Update 11am: Leo Varadkar says it is important for the peace process that the British Government is not too close to any political party in the North.
The Social Protection Minister is expected to be elected Taoiseach on Wednesday.
His comments come as talks to restore powersharing are resuming at Stormont today.
Sitting Taoiseach Enda Kenny raised his concerns with Prime Minister Theresa May.
The British Conservatives have already agreed a deal in principal with the DUP for the government in Westminster.
But Minister Varadkar says both governments must be co-guarantors of the Good Friday Agreement.
"When I have an opportunity to speak to Prime Minister May I'll emphasise that as well, because our role as governments, here in Dublin and in London, is to act as co-guarantors and not to be too close to any particular party in the North, whether it's a nationalist or republican party or a unionist party and that is certainly something that I will emphasise in any context that I have with Prime Minister May," he said.
Earlier: The leader of the Democratic Unionists has pledged to exercise the party's enhanced influence at Westminster responsibly.
Arlene Foster said she hoped to capitalise on opportunities the situation presented for the North.
As talks continue between the DUP and Conservatives over a potential confidence-and-supply arrangement to support Theresa May's minority government, Mrs Foster said: "When I meet with the Prime Minister in London tomorrow, I will be mindful of our responsibility to help bring stability to the nation at this time of challenge.
"We will be working to agree arrangements that can provide the whole nation with good government.
"The DUP will work to bring about outcomes that are beneficial to all, and in Parliament Northern Ireland's case will be centre stage."
It is likely the DUP will press for increased investment in the North as the price of their support in Westminster and push for a more significant role in the Brexit process.
The DUP's social conservatism on issues such as gay marriage and abortion has been in the spotlight in Great Britain since its role as parliamentary kingmaker became clear.
Mrs Foster branded some of the commentary and analysis about her party as "inaccurate and misleading".
"I have no doubt over time those responsible will look foolish in the extreme," she said.
Mrs Foster, writing in the Belfast Telegraph, said the election result had caused a "political earthquake" across the United Kingdom.
"In truth, no-one expected the outcome of the snap general election to be a hung Parliament, and for the DUP to be in such an influential position," she said.
She added: "The mandate given to us by the people will be used responsibly.
"We stood on a clear policy platform of wanting to strengthen the Union, of working for a good deal for Northern Ireland as the United Kingdom leaves the EU and of promising to do our best to get Stormont up and running again for the benefit of all.
"We will use the position we find ourselves in to do as we promised."
Mrs Foster concluded: "The next few weeks represent a real opportunity for everyone in Northern Ireland to heed the will of the people and capitalise on the opportunities that lie ahead for everyone."