Northern Ireland's former First Minister Arlene Foster has said she is open to talks with Sinn Féin to avert a meltdown of power-sharing institutions.

The Democratic Unionist leader also announced plans for a public inquiry into the botched green energy scheme that prompted the resignation of Deputy First Minister Martin McGuinness yesterday.

She insisted the inquiry could go ahead without the sign-off of the DUP's partners in government, Sinn Féin.

"It's needed to restore confidence in the institutions and also for me personally, to retain my integrity, which has been completely maligned over this past number of weeks and months."

"We are willing to take part with any discussion to see if a way forward can be found," she said at a news conference.

"I remain open to further discussions with Sinn Féin or any of the other parties in the Assembly over the next few days."

The departure of Sinn Féin veteran Mr McGuinness amid a row over the Renewable Heat Incentive (RHI) forced Ms Foster from her job as First Minister as well.

Theoretically the parties have seven days to resolve their differences before Northern Ireland Secretary James Brokenshire has to call a snap poll.

However, Mr McGuinness has made clear there will be no going back to the status quo and his party is preparing to face the electorate.

Ms Foster said a DUP minister would announce plans for a public inquiry into the RHI affair later this week.

The furore has left Stormont facing a £490m overspend.

She said it was important for Stormont's reputation and her own.

"It's needed to restore confidence in the institutions and also for me personally, to retain my integrity, which has been completely maligned over this past number of weeks and months," she said.

Earlier, the Secretary of State for Northern Ireland told MPs that the situation in Northern Ireland is "grave" and that early elections look likely.

Mr Brokenshire this afternoon updated the House of Commons following Mr McGuinness's resignation.

He said that "while the Renewable Heating Incentive might have been the catalyst, it has however exposed deeper tensions between the parties".

He said an impartial inquiry into the RHI scheme needs to be established as quickly as possible.

Mr Brokenshire urged both sides to try to find a way forward, and said both the British and Irish governments will offer their support.

However, the Secretary of State said "the clock is ticking" and if there is no resolution he will have to call an election.

Northern Ireland is now in its longest period of stability and that should not be lightly thrown away, he said.

He will do all he can to establish a way forward, to ensure a bright and prosperous Northern Ireland, he added.

Taoiseach and British Prime Minister discuss crisis

Meanwhile, Taoiseach Enda Kenny and British Prime Minister Theresa May spoke on the phone for around15 minutes this evening to discuss the latest developments in Northern Ireland.

They agreed that the situation is very serious and that the two Governments would work closely over the coming period.

In particular, Minister for Foreign Affairs Charlie Flanagan and Mr Brokenshire will work over the next few days to see if a way forward can be found before an election has to be triggered.

Mr Kenny and Ms May agreed to maintain close contact ahead of Ms May's planned visit to Dublin at the end of January.

No way to avoid election - Adams

Mr McGuinness's decision to walk away after ten years sharing power with the DUP came as Ms Foster refused to stand aside to facilitate an inquiry into the ill-fated RHI - the so-called "cash for ash" - scheme.

".....you can be sure that him coming to this decision, with our full support, was done reluctantly. It was the last thing that Martin wanted to do."

Sinn Féin President Gerry Adams this afternoon said that he does not think there is any way to avoid holding an election in Northern Ireland.

Mr Adams said that Ms Foster's refusal to step aside left Mr McGuinness in an untenable position, adding that he would not have made the decision to resign lightly and that he would decide over the next number of weeks if he would lead the party into the expected election.

Speaking on RTÉ's News at One, he said: "The position was absolutely untenable. He's invested ten years. He worked with Ian Paisley when people said that wasn't possible. He worked with Peter Robinson. He's a tireless worker. He's reached out and discommoded Republicans at time with his outreach.

"So you can be sure that him coming to this decision, with our full support, was done reluctantly. It was the last thing that Martin wanted to do."

Mr Adams said the only way to get the power-sharing executive back in place was if there was a commitment to the principles on which the Good Friday Agreement was embedded.

He said people would vote against corruption and for the key elements of the agreement, which were fairness and equality.

Mr Adams said Mr McGuinness and Mr Paisley managed to make power sharing work but that Ms Foster had failed because she did not acknowledge all the people of the North.

Alliance MLA David Ford has said that an election was not necessary, but is now inevitable.

He told RTÉ's Drivetime that the announcement of an inquiry today into the botched green energy scheme is a month too late.

He said the power sharing executive between the DUP and Sinn Féin reached a point where there was no trust and no willingness to seek to find a way through the problems.

He said Northern Ireland's former First Minister, Arlene Foster has chosen to show that she bore no responsibility for the scheme and this attitude, he said, was where the issue of the RHI became a problem for relationships between the two parties.

'Utter abdication of responsibility', says Dodd

DUP deputy leader Nigel Dodds has said that it is with deep regret that Northern Ireland is facing fresh elections, adding that it is not in his party's making, and describing it as an "utter abdication of responsibility".

Nigel Dodds, DUP deputy leader says the party is 'deeply, deeply saddened' that Northern Ireland has been plunged into a 'political crisis' pic.twitter.com/9S3VtNG6Su — RTÉ News (@rtenews) January 10, 2017

He told RTÉ's Six One that Sinn Féin has decided to "toss the toys out of the pram" as far as power-sharing in Northern Ireland is concerned.

The DUP MP said that his party is deeply saddened that Sinn Féin has plunged Northern Ireland into a political crisis and into a needless election.

He said Sinn Féin has decided to tear down the institutions and accused the party of now "doing the wrecking" in Northern Ireland and said it is not about the RHI scheme.

"There are obviously issues that they have to address internally ... It will be very, very difficult after a bruising election to come back and put everything together again."

While politicians are set to face the electorate, a poll is unlikely to resolve the crisis if the RHI issue is not dealt with before an executive is re-formed.

The state-funded RHI was supposed to offer a proportion of the cost businesses had to pay to run eco-friendly boilers, but the subsidy tariffs were set too high and, without a cap, it ended up paying out significantly more than the price of fuel.

This enabled applicants to "burn to earn" - getting free heat and making a profit as they did so.

Claims of widespread abuse include a farmer allegedly set to pocket around £1m in the next two decades for heating an empty shed.

While the DUP and Sinn Féin were in agreement on the terms of a potential investigation into RHI, the sticking point was the position of Ms Foster when the probe got under way.

Steps by the Executive to cut the costs of the overspend will not be implemented in the short term.

Mr McGuinness cited other disputes with the DUP, including over the Irish language and stalled mechanisms to deal with the legacy of the Troubles, in explaining his move.

The DUP claimed RHI was not the motivation behind Sinn Féin's strategy, insisting they were exploiting the crisis to pursue a broader republican agenda.