Mrs Foster said gay rights were social issues for the Northern Irish parliament to deal with.
Sinn Fein's Stormont leader Michelle O'Neill said: "I will be making it very clear that any deal between the Tories and the DUP can not be allowed to undermine the Good Friday and subsequent agreements".
The British prime minister is set to travel to Paris to hold talks with French President Emmanuel Macron after the meetings today.
Pic: ReutersGerman Chancellor Angela Merkel said on Wednesday the European Union was ready to start negotiating with Britain about its exit from the EU on June 19, as planned, despite surprising election losses for London's governing Conservatives.
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.
Same-sex marriage campaigners in Northern Ireland have urged thousands of people to take to the streets in protest against the DUP.
During his time in office, Major helped to start the process that eventually led to the end of violence and a finely balanced local assembly in Belfast - where power is shared between the mainly Protestant unionist community, which celebrates its British heritage, and the mainly Catholic nationalist community, which identifies as Irish.
Blimp crashes at US Open
Golfer Brandt Snedeker said he spotted something while on the course at Erin Hills, about 35 miles northwest of Milwaukee. Bystanders shared shocking videos of the blimp catching on fire before hitting the ground near highways 83 and 167.
No-one is clear what the UK wants any more: the hung parliament election result has left May's plans for a clean break with Europe, under a so-called "hard Brexit", in tatters as her emboldened rivals in the UK Parliament demand it is watered down.
Sinn Féin Leader in the North Michelle O'Neill has said that regardless of talks between the DUP and Tories all roads must lead back to an Executive, which delivers for all.
Meanwhile, the chief European Union negotiator has told the Financial Times that the clock is ticking on Brexit talks, and that Britain should be wary of further delays.
The Irish republican Sinn Fein party - which won seven seats in the election although their MPs traditionally do not take up their seats in protest - is also wary of the alliance. "I can't negotiate with myself", he told European newspapers including the Financial Times.
That concern is made more pressing by two factors: the looming threat of Brexit, which could make it necessary to once again erect barriers along the now invisible border between Northern Ireland and Ireland, once the United Kingdom leaves the European Union; and the fact that Northern Ireland's local assembly is in crisis.
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.
Ms May faces a hard balancing act: Divisions over Europe helped sink the premierships of Margaret Thatcher, John Major and David Cameron, and many of her politicians and party membership support a sharp break with the EU.
The performance of the British economy could also influence perceptions of Brexit.