"Regrettably, thus far in the talks process there has been no substantive progress across all of the key issues that are at the core of the current impasse", Sinn Fein president Gerry Adams said.
A snap election on March 2 led to Sinn Féin, which wants a united Ireland, winning 27 seats, one fewer than the Democratic Unionist Party, the largest in the Executive.
He said that despite the British government's efforts, "agreement at this stage has not proved possible" as there were "significant gaps" between the Democratic Unionist Party and Irish nationalists Sinn Fein.
DUP MLA Arlene Foster, who was First Minister before the Assembly was dissolved in January, has said: "This suggestion that every single person who's a homosexual wants to change the definition of marriage is actually wrong".
Sinn Féin has previously said it would not agree to Foster being First Minister until an inquiry into the "cash for ash" controversy has concluded.
The failure to achieve a deal ahead of yesterday's 4pm deadline left Brokenshire with three options: a return to direct rule from Westminster, another snap election, or granting the parties more time for talks.
"Everyone owes it to the people of Northern Ireland to grasp that and provide the political leadership and the stability they want".
During parliamentary exchanges following his statement, Mr Brokenshire faced repeated questions from the opposition benches on why Mrs May had not directly intervened in the process.
"From this House, we must ensure that the hands-off, let them get on with it, it's all done and dusted attitude that prevailed.is ended".
Masters co-leader Sergio Garcia ready to change major fortunes
They had a one-shot lead over Rickie Fowler, who made birdies on all four of the par 5s to offset a few mistakes in a round of 71. So he's already shown some bad signs, and that was without the pressure of going for his first major win during the final round.
Mr Brokenshire reiterated his view that the situation was "not sustainable" in the long term.
"It is actually very serious and the fact that we do not have an executive being formed today is very regrettable".
"We now have a short window of opportunity to resolve outstanding issues and for an executive to be formed", he said.
The discussions have also been keenly followed by Dublin, with Irish Foreign Minister Charlie Flanagan calling for power-sharing to be re-established.
The BBC reports in a staff memo, the head of the Northern Ireland Civil Service Sir Malcolm McKibbin has promised to try to maintain a business as usual approach during what he describes as "this time of uncertainty".
But the two parties have failed to reach an agreement since the election - with each blaming the other for this.
Mr Brokenshire said today that Northern Ireland was approahing the point where the government will "not have an agreed budget" which could have knock on effects for public services.
Proposed legislation to protect Irish language speakers and new mechanisms to deal with the legacy of the Troubles remain logjams in the way of consensus.
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