It comes just two days after Theresa May and Leo Varadkar emerged from meetings in Belfast saying they were hopeful of a deal this week.
Disagreement has focused on the demand from nationalist parties for an Irish language act, which the DUP has said it “will not countenance”.
“Despite our best efforts, serious and significant gaps remain between ourselves and Sinn Fein especially on the issue of the Irish language.
I have made it consistently clear that unionists will not countenance a stand alone or freestanding Irish Language Act. Sinn Fein’s insistence on a stand alone Irish Language Act means that we have reached an impasse.
“As far back as last summer, I outlined my party’s willingness to reach an accommodation on language and cultural issues. However, I indicated that any such accommodation must be fair, balanced and capable of commanding support on all sides of our community. At the moment, we do not have a fair and balanced package.”
Power sharing at Stormont has been suspended for the past 13 months, despite two elections and intensive talks seeking to bridge the gap between the DUP and Sinn Fein.
Failure to reach agreement means the imposition of direct rule from London appears inevitable, with the DUP calling on the UK Government to pass a budget for Northern Ireland.
Ms Foster added: “After the Assembly election, I embarked on an engagement exercise with those who love and cherish the Irish language. I respect the Irish language and those who speak it but in a shared society this cannot be a one-way street. Respect for the unionist and British identity has not been reciprocated.
“In our view, there is no current prospect of these discussions leading to an executive being formed.
“It is now incumbent upon Her Majesty’s Government to set a budget and start making policy decisions about our schools, hospitals and infrastructure. “Important decisions impacting on everyone in Northern Ireland have been sitting in limbo for too long.”
The DUP leader concluded: “Restoring a sustainable and fully functioning devolved government will remain our goal but we will not accept a one-sided deal.
“Any agreement to restore the executive must be on a sensible basis. We cannot and will not be held to ransom by those who have refused to form an executive for over thirteen months."