"What I am actually advocating is unionism taking a stand now, saying that "look, enough is enough", he said.
Ms Foster said she hoped power-sharing could be restored but had concluded "that is not possible at this time".
But she blamed the DUP for failing to "close the deal".
The absence of an executive has limited Belfast's say in Britain's negotiations to leave the European Union, which are set to have a bigger impact on Northern Ireland than on any other part of the United Kingdom.
The Dublin government has signalled that it will be seeking direct input into how Northern Ireland is governed following the DUP's decision to halt the latest effort to restore devolution. "These issues are not going away", she added.
Mary Lou McDonald, the new Sinn Fein leader, said that the two sides were close to an agreement.
She said "the basis for an accommodation still exists" but, raising the prospect of direct rule being reintroduced, said "challenging decisions" would now have to be taken by the United Kingdom government.
SDLP Leader Colum Eastwood. Mandatory Credit Stephen Hamilton
While both parties have acknowledged that progress had been made, a DUP source has played down expectations of a deal being finalised on Monday, suggesting to the Press Association that if an agreement does materialise it is more likely later in the week.
In the almost 20 years since the 1998 Good Friday Agreements brought peace to Northern Ireland, the long-time political leader of Ulster's Republican movement Gerry Adams is stepping aside after his successor was elected at a party congress on February 10.
Officials from Dublin and Westminster will work together on efforts to avoid a hard border between the United Kingdom and Ireland after Brexit, Leo Varadkar said.
THERESA May has been left humiliated by her own DUP partners today as a deal to form a Government in Northern Ireland broke down just 48 hours after she flew to Belfast. "The DUP should reflect on their position".
Proposals for dealing with Northern Ireland's violent past involving extra money for historic investigations and truth-telling processes have also been delayed by the impasse and may fall to Westminster to implement.
"The SDLP leader should be less concerned about Theresa May's feelings and more concerned about getting an agreement that enables genuine power-sharing based on rights, respect and equality for all".
After decades of trying to find a way for our communities to work together, we are retreating again.
Northern Ireland has been without a functioning government since January 2017, with two elections and several rounds of talks failing to resolve the crisis.