Ian Paisley made significant gains in the election
Northern Ireland Secretary Paul Murphy is meeting the main parties in the newly-elected assembly to discuss the future of devolved government.
The Democratic Unionist Party (DUP) which opposes the Good Friday Agreement, overtook the Ulster Unionists to become the biggest party.
However, the DUP refuses to share power with Sinn Fein, which also made gains.
The assembly was suspended more than a year ago and the parties went into the election against the background of a deadlocked political process.
Sinn Fein President Gerry Adams said his party would be pressing the government to lift the suspension and get the institutions up and running again.
"We will use our mandate wisely. We won't wait until the DUP get themselves into better shape.
"We're going to press the governments to move ahead, and we're going to meet the other parties," he told a news conference in Belfast.
Earlier, DUP spokesman Ian Paisley junior said the Good Friday Agreement was "dead in the water" and his party was demanding a new deal.
Paul Murphy: Holding talks with main parties
The Northern Ireland secretary is meeting the SDLP, Sinn Fein and Ulster Unionists separately on Saturday to examine the way forward.
He said the parties would have to work together.
Ulster Unionist leader David Trimble has predicted a deadlock for "the next few months" but said it was a "huge overstatement" to say the Agreement was dead.
"There is still a majority of the population in favour of the Agreement," he told BBC Radio 4's Today programme.
The British and Irish Governments have insisted that the Agreement remains the only viable political framework - and was not open to negotiation.
They promised to bring forward proposals in the new year for a review of the Good Friday Agreement.
Mandate
The BBC's Ireland correspondent Mark Simpson said there could be little movement until DUP leader Ian Paisley and Gerry Adams of Sinn Fein got together to talk.
But Mr Paisley maintains his refusal at "all costs".
"He wouldn't even share a cup of tea with him, never mind share power," our correspondent said.
Despite the governments' firm line, the DUP said the election - in which it won 30 seats - had given it a mandate for renegotiation.
Ian Paisley junior - who won a seat alongside his father in North Antrim - told BBC Radio 4's Today programme that it was time the British Government "wakens up to the reality" that a new deal had to be sought.
"It's dead in the water. The Agreement is over - that was the message of this election," he said.
The DUP is expected to meet Mr Murphy early next week.
However, Mr Murphy said the fundamentals of the Agreement - such as the principles of power-sharing and consent of the people - could not be changed.
'Crisis within unionism'
"Northern Ireland can only be governed by an accommodation between nationalists and unionists, and that accommodation over the last five or six years has been hugely successful," he said on Saturday.
"I am not underestimating the difficulties, but I am not unhopeful that we can make progress," he added, saying that power-sharing between the hard liners had already happened "whether they talked to each other or not".
Sinn Fein secured 24 seats in comparison with the SDLP's 18 - a direct reversal of the parties' positions after the last election.
The Alliance gained six assembly places, while the three remaining seats went to a County Tyrone doctor standing on a single issue over hospital services, maverick unionist Robert McCartney and Progressive Unionist David Ervine.
Sinn Fein President Gerry Adams said it had been a good election for his party.
Reflecting on his party's showing, SDLP leader Mark Durkan said it had to "work with the hand that democracy deals us".
The White House admitted it had some concerns over the outcome of the election.
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The process is dead in the water
