Northern Secretary Teresa Villiers does not believe “business as usual” is possible at Stormont as the row over the Provisional IRA continues, according to DUP deputy leader Nigel Dodds.

Speaking after his party’s meeting with Ms Villiers at Stormont House, Mr Dodds said the situation in Northern Ireland had been “profoundly difficult” since the Chief Constable George Hamilton said IRA members may be involved in the killing of Kevin McGuigan.

“For us, the basis of entry into Government was that people were committed to exclusively peaceful and democratic method of operation,” Mr Dodds said.

“That remains the basis on which government has to continue in Northern Ireland. We raised with the Secretary of State that it cannot be business as usual until this matter is resolved and she agreed with us.”

Describing the meeting with Ms Villiers as constructive, the deputy leader of the North’s largest Unionist party said he intended to “keep the pressure” on Sinn Féin. “We’re not going to let them off the hook. There are very serious issues that they have to address. Up to now they seem to be running away from those questions. We don’t intend to let them run away from them.”

He said his party was determined that, “one way or another”, there would be an administration in Northern Ireland “consisting of people totally committed to peaceful and democratic means only. If other parties do not step up to the plate with us, or if the Government does not take sufficient action to deal with this matter, then we will bring about the circumstances to create the time and space in which this matter will be resolved.”

Mr Dodds said if Sinn Féin did not deal with the situation his party would move to have them excluded and devolved government would not continue on that basis.

He said the DUP had asked for a meeting with British prime minister David Cameron and was he was hopeful it would take place “very soon”.

He said his party had also raised with Ms Villiers the issue of the “revocation of licences” of people who have been convicted of very serious crimes, such as murder, and were out of prison on licence.

“We urged the Secretary of State along with the police to give the speediest consideration to the revocation of licences of anyone involved in and around the murder of Kevin McGuigan and indeed other crimes as well.”

Mr Dodds said there was “sufficient basis” to put down a motion to exclude Sinn Féin from the Executive. Cross-party support would be required for such a motion to succeed.

“There is no reason at all, as things stand at the moment, why there should not be the exclusion of Sinn Féin on the basis of what the Chief Constable has already said.”

The Executive is a power-sharing arrangement involving ministers from the five biggest parties in the Assembly.

On Wednesday, the Ulster Unionist Party (UUP) said it intended to leave the Executive, putting pressure on the DUP to act.

Earlier on Thursday, DUP MP Jeffrey Donaldson stated his party will seek to exclude Sinn Féin from the Executive.

“We are going to move to exclude Sinn Féin from the Government,” the Lagan Valley MP told BBC Radio 4.

“In the end, if the other parties are not prepared to support the exclusion of Sinn Féin, then we will act unilaterally, and if that means that we have a period in Northern Ireland where we don’t have a government until we resolve and sort out these issues then so be it,” added Mr Donaldson.

Mr Donaldson did not say the DUP would follow the UUP out of Stormont but he said Sinn Féin needed to deal “with this issue in terms that gives everyone the reassurance that this isn’t going to happen again”.

Mr Donaldson suggested introducing a monitoring arrangement, similar to the former Independent Monitoring Commission, to ensure the IRA is not engaging in violence. “If the IRA exists in a form where its members have access to weapons and are prepared to go out and kill people, then that is not acceptable,” he told RTÉ radio’s Morning Ireland.

However, Fianna Fáil leader Micheál Martin disagreed with the suggestion that Sinn Féin be excluded, stating it “is not the way to go”.

“I’ve invested a lot in the peace process and my party have as well. The peace process belongs to the nation, it’s not any particular party’s peace process. It’s not Sinn Feein’s peace process. It belongs to the people.

Sinn Féin finance spokesman Pearse Doherty said, despite claims to the contrary, he was confident the structures of the IRA had gone away.

Asked at a press conference in Dublin how he knew this, Mr Doherty said: “We represent communities in the North which gave life to the IRA, which sustained the IRA for over 30 years in the armed conflict. We know what is happening in those communities.”

Mr Doherty said there were members of the IRA who still are in contact with each other, but insisted the organisation had been stood down. "Former volunteers still exist, they didn't all pass away, they didn't all leave the island, they are still there involved in Sinn Fein, some of them are involved in the GAA, some of them are involved in community groups, some getting on with their lives.

“I am sure on occasions they sit down and have a drink or meet at an event, but is that a meeting of the IRA? No it is not, the IRA have gone, they have left the stage, they have disbanded, they have out the weapons out of use and that happened 10 years ago.”

Last night, former taoiseach Bertie Ahern said it was “time for cool heads” and a calm focus on the peace process to ensure the Stormont institutions remain standing.

“The bigger picture is the ongoing peace process,” said Mr Ahern. “It is always delicate. Cool nerves . . . think this through. It is terrible, there are atrocities, but are those atrocities being done who might have been on the inside but are now long since on the outside?”

He warned of the dangers of political collapse. “There is a small dangerous group who does not want to see the spirit or the progress of the Good Friday agreement implemented,” said Mr Ahern. “Nothing should be done that creates a vacuum that allows them to expand and grow.”

Following apparent differences between the PSNI and the Garda over whether Provisional IRA structures continue to exist, the Garda Commissioner Noírín O’Sullivan issued her first statement on the matter.

She said the Garda has never denied the Provisional IRA’s existence, but it had no intelligence or information that it “still maintains its military structure”.

She said the Garda keeps its security assessments under constant review.