The governments have urged Northern Ireland's parties to reach agreement on the date for devolution after power was restored to the assembly.
Martin McGuinness (left) would be deputy first minister
It comes amid speculation that DUP leader Ian Paisley will hold his first formal talks with Gerry Adams.
Such a meeting would represent a significant policy shift away from the DUP's long held official policy of no contact with Sinn Fein.
An order signed by the NI secretary restarted devolution at midnight.
However, Peter Hain said Monday's deadline for a power-sharing executive to be formed must be met.
The Northern Ireland secretary says the assembly will be dissolved if it is not - but he will consider an alternative if the parties can agree on one.
The DUP says it will enter government with Sinn Fein, but not until May.
Ulster Unionist leader Sir Reg Empey said Mr Paisley was putting Mr Adams into a strong negotiating position.
"What is happening here is that Ian Paisley is preparing - and the trajectory of where he is going is quite clear - he is going into power with Sinn Fein," he said.
"Whether he does it today, or in a few weeks' time, the government will probably argue it is neither here nor there," he told the BBC on Monday.
The SDLP said Mr Hain had "caved in to the DUP".
Reacting to the DUP's statement on Saturday, Sinn Fein accused the DUP of trying to "frustrate the will of the electorate".
The DUP's Lagan Valley MP Jeffrey Donaldson said that his party agreeing to rule with Sinn Fein was an "historic decision".
"The Democratic Unionist Party has made a firm commitment to participate in a power-sharing government in May," he said.
"That is absolutely clear and unequivocal - we felt there was a need for more time because there were outstanding issues that need to be resolved."
Mr Hain insisted Stormont would be dissolved if Monday's deadline was not met - but said if parties could agree an alternative, he was open to that.
The government has threatened to dissolve Stormont
The British and Irish governments say they will shut the assembly and stop the pay of its members if a power-sharing executive is not agreed on Monday.
If devolution does not return, controversial water bills will also be posted to homes in Northern Ireland within days.
Chancellor Gordon Brown has promised an extra £1bn if devolution is back on Monday on top of £35bn promised by the government over four years.
Speaking on Sunday, Mr Hain said: "Success tomorrow would be devolution, failure would be dissolution. I don't at the moment see any other way, I only see it our way.
"If there is another way if the parties have got their own way then they need to jointly agree it and come back to me pretty quickly, because otherwise the law kicks in and there's nothing I can do about it."
On Saturday, the DUP said it was willing to bridge the gap between now and May with preparatory work, "including departmental pre-briefings and finalising a programme for government".
It said it wanted the extra time "to raise the level of confidence in the community and instil a positive attitude towards devolution and local control".
If nominations take place the DUP is expected to put forward party leader Ian Paisley as first minister, with Sinn Fein choosing their chief negotiator Martin McGuinness as his deputy.
If a power-sharing executive is formed, it will have four DUP ministers, three Sinn Fein, two UUP and one SDLP.
The Northern Ireland Assembly has been suspended since October 2002, amid allegations of an IRA spy ring at Stormont.
A subsequent court case collapsed. Direct rule has been in place since that date.