Sinn Féin is to hold separate meetings with the prime minister and taoiseach (Irish prime minister) next week.
Party leader Mary Lou McDonald said she will be calling on the two governments to convene the British-Irish Intergovernmental Conference to avoid a prolonged stalemate.
Both Sinn Féin and the DUP had been engaging in negotiations in a bid to end the 13-month impasse at Stormont.
Speaking in Dublin, Ms McDonald said the party still believes a restoration of devolution is possible and that direct rule is "not an option".
She said: "We will be telling the two governments that now is not a time to stand still. Issues are still outstanding and they have to be resolved".
Ms McDonald also appealed to the DUP to return to the talks.
On Wednesday Mrs Foster said there was "no current prospect" of a deal to restore power sharing in Northern Ireland.
She said talks failed due to disagreements with Sinn Féin about legislation for the Irish language.