The reports were seen as a sign that the party's Scottish leader was seeking to assert the influence of her MPs.
Ms Davidson was elected as leader of the Scottish Conservatives in 2011 after opposing a similar plan put forward by rival Murdo Fraser, now the party's finance spokesman.
Ms Davidson responded by tweeting: "Folk might remember I fought a leadership campaign on the other side of that particular argument".
Davidson, who led the Scottish Conservatives to its best performance since 1983, said she supported Prime Minister Theresa May who has said she will form a new government with assistance from Northern Irish unionists and press on with talks with the European Union to secure a successful Brexit deal.
The Scottish Daily Telegraph's editor Alan Cochrane said: "The story was impeccably sourced and I stand by every word".
Asked if Scottish Tories would push for a softer line on Brexit, he said: "I think we can have a tremendous amount of leverage".
"I asked for a categoric assurance that if any deal was done with the DUP there would be absolutely no rescinding of LGBTI rights in the rest of the United Kingdom, and that we would try to use any influence that we had to advance LGBTI rights in Northern Ireland".
This morning, Ed Vaizey, the former Conservative culture minister, also insisted the Conservative party would remain socially liberal and not bend to the social conservatism of the DUP.
The DUP opposes same-sex marriage and Northern Ireland is the only part of the United Kingdom where it is not legal.
In November 2015, the Northern Ireland Assembly voted in favour of same-sex marriage, but the DUP blocked the motion after controversially stating there was a "petition of concern".
Ms Davidson said: "I was fairly straightforward with her and I told her that there were a number of things that count to me more than party".
"Hard Brexiters have got some thinking to do - a minority government with the stresses and strains of coalition - may be a short-lived affair", he added.
"One of them is country, one of the others is LGBTI rights".