The Prime Minister and Taoiseach will travel to Northern Ireland later amid growing speculation that a deal to restore the powersharing government is imminent.
Mrs Foster released a statement on Wednesday afternoon in which she called for the Westminster government to set a budget. The Prime Minister said after the meeting, "while some differences remain, I believe it is possible to see the basis of an agreement".
She added: "In our view, there is no current prospect of these discussions leading to an Executive being formed".
"What we are trying to find is an accommodation and a way forward that values those people who are Irish speakers but doesn't impinge on the lives of those who aren't Irish speakers and I think that's important".
Since then, divisions over issues including Irish language rights, same-sex marriage and how to deal with the legacy of Northern Ireland's troubled past have proved insurmountable.
The leader of Alliance Party, Naomi Long, said: "We are in a very precarious situation at this point in time, we are essentially in unchartered territory".
"We can not and will not be held to ransom by those who have refused to form an Executive for over 13 months".
Irish Prime Minister Leo Varadkar said he is "very hopeful" of an agreement being reached this week.
Foster described the deal proposed by Sinn Fein as unfair and unbalanced. They can not get close to solving them if Sinn Fein and the DUP won't even speak to one another.
"It is now clearer than ever that the Petition of Concern must be reformed - up front as part of any Stormont Talks deal - if the devolved institutions are to have any credibility in being able to provide a government for everyone".
"We should allow the majority of the Assembly's voice to be heard and the majority of the community's voice to be heard in making sure that everybody on this island is entitled to have their love fully respected and be married if they so wish", he said.
Irish senator Neal Richmond said repeated warnings against any type of border between Northern Ireland and the Republic were not "alarmism".
Traditional Unionist Voice (TUV) leader Jim Allister, an arch critic of the mandatory coalition government, said it was an overdue conclusion to a "pointless talks process".
The government fell in January a year ago over a series of issues, including a scandal surrounding a renewable heat scheme and the departure of deputy first minister Martin McGuinness, the nationalist Sinn Fein's main figure, due to ill health.