British Prime Minister Theresa May has said there will be no question of Britain sharing joint authority with Ireland in the event of direct rule being reintroduced in Northern Ireland.

Answering a question from the DUP's Nigel Dodds during the first Prime Minister's Questions since the UK Parliament's summer recess, Mrs May said talks to restore a devolved administration in Northern Ireland were important.

She added: "I am happy to confirm that we will not be looking at a joint authority.

"He [Mr Dodds] will be aware that the Belfast Agreement does include within it certain responsibilities in relation to the Government of the Republic of Ireland in North-South coordination, but I think that the focus for all of us should be on trying to ensure that we can resolve the current differences and we can see that devolved administration reasserted in Northern Ireland.

"I think that is what would be best for the people of Northern Ireland."

Minister for Foreign Affairs Simon Coveney had earlier said direct rule from London could not be imposed without input from the Irish Government.

Mr Coveney said there were still grounds for optimism over the preliminary power-sharing talks in Belfast and direct rule should be avoided.

"There can be no British-only direct rule. That is the Irish Government's position," Mr Coveney said.

"It would be very difficult to even contemplate how direct rule would function in that context."

Meanwhile, Sinn Féin's leader in Northern Ireland Michelle O'Neill has said there is now the possibility of progress being made to soon restore the political institutions in Stormont.

Ms O'Neill said the Sinn Féin and DUP leaderships have held intensive private talks for more than a week and that formal talks with all parties should now begin.

She said the path to restoring the Northern Executive is clear and it could be achieved through short and sharp negotiations.

Those talks will have to resolve outstanding issues such as the Irish Language Act and marriage equality.

Speaking at a meeting of her party's elected representatives in Co Meath, Ms O'Neill also accused Fianna Fáil and Fine Gael of arrogance in claiming Sinn Féin were not fit for Government in Dublin.

In her speech she said: "We do believe progress is possible and are therefore ready to re-engage in formal negotiations together, and with the other parties and both Governments, to try and reach agreement in a short, sharp and focused negotiation."

Her remarks will increase speculation that the British and Irish governments plan to call formal talks with the five main parties in the coming days.

DUP leader Arlene Foster has confirmed that her party has been involved in what she calls "detailed engagement" with Sinn Féin to see if a basis can be found to restore power-sharing. 

In a statement released this morning, Mrs Foster said: "At the end of the formal talks process at Stormont Castle, I indicated that we would continue discussions with Sinn Féin as well as the other parties.

"We have stepped up that engagement with Sinn Féin over the last week during which time we spent a number of days involved in a detailed engagement with them to ascertain whether an agreement on the issues before us is possible.

"We intend to continue with a further series of bi-laterals with all of the other parties to determine whether agreement can be reached in the short time available."

Wilson rejects Adams' unity referendum claims

Elsewhere, a DUP MP has said he believes Sinn Féin President Gerry Adams is stringing people along by calling for a referendum on Irish unity.

Speaking at a party think-in yesterday, Mr Adams said he wanted such a vote within five years, which he said would be winnable.

Speaking on RTÉ's Morning Ireland, Sammy Wilson said Mr Adams was effectively holding a carrot out in front of the donkey because he had to give his party some hope that the failed strategy of abstentionism, of irresponsible behaviour in government, will eventually lead to some positive outcome for them.

Mr Wilson said a referendum will only be called if and when the Northern Secretary deems that there is a chance of the people voting for a change in the status of Northern Ireland.

"There is no indication of that," he said.

He added that "opinion polls show that many nationalists, while they'll vote for nationalist parties, will not vote for Irish unity".

Mr Adams, also speaking on Morning Ireland, said the Good Friday Agreement makes it clear that there can be a referendum on Irish unity and his party wants that.

He said Sinn Féin wants "an agreed Ireland" and it wants "a debate on that".

He also called the establishment of "an Oireachtas Committee to look at what shape Irish unity would take".

He added: "Those who want the union have the right to argue that, to debate that to win us over to that notion.

"The perpetual majority, which the unionists use to enjoy and for which the Northern State was actually created has gone. 

"They no longer have an electoral majority so they have to look at other ideas of how we shape out our future."