Sinn Féin leader Mary Lou McDonald has called for the Irish and British governments "to step in urgently and map out the answers" after rejecting a DUP proposal to hold talks on contentious issues separately to agreeing on the resumption of power-sharing.

Ms McDonald said the only way to get sustainable governance in Northern Ireland was "to hardwire equality" by extending same-sex marriage and Irish language rights.

Earlier, DUP leader Arlene Foster said it was important that the Northern Ireland Assembly gets back up and running along with a parallel process to "deal with the issues Sinn Féin wants to raise in relation to other matters".

Ms Foster told RTÉ's Morning Ireland that her proposal offered a balanced way forward after the acrimony of last year.

Speaking to RTÉ's Today with Miriam, Ms McDonald said that these issues of equality are not just Sinn Féin issues, but issues that were important to a wider community.

She said it was inaccurate to say Sinn Féin had not engaged or compromised, as it put an accommodation on the table last February, which was "fair and balanced ... unfortunately the DUP walked away from it".

The two party leaders were speaking a day after sitting beside each other at the funeral of murdered journalist Lyra McKee.

Ms McDonald said that Fr Martin Magill had laid down a very serious challenge "not just to talk and tick boxes but to get the Assembly up and running again".

She said Sinn Féin agrees and is ready to talk and engage but "would not capitulate to people who wish to hold back progress in every form and who don't wish to make room for others" in a democratic society.

Ms McDonald said that Fr Magill "hit the nail on the head" in the clearest and most uncompromising way that politicians, including Ms McDonald, need to roll up their sleeves and "get cracking".

"If we have the political will we can sort these issues out in a matter of hours because we have done the talking and what we need is the decision, the resolution," she said.

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Ms Foster said she believes there is a great desire to see the Stormont Assembly up and running - and it is important to do that - but said she does recognise the need for a talks process alongside that.

She said there is a willingness to talk and if it means holding other talks first, the DUP will be there too.

Ms Foster said Fr Magill spoke very powerfully and plainly about what people want and said it is clear they want to see devolution back and power-sharing back in Stormont.

She said that Ms McKee's funeral was a very moving service and we must not forget it was about the brutal death of a young woman cut down in the prime of her life.

Ms Foster said the violence that led to Ms McKee’s death was to further those responsible’s "own warped political agenda", not because of the political vacuum that exists in Northern Ireland.

Restoration of NI power-sharing will involve compromise - Methodist minister

Efforts to restore power-sharing at Stormont will involve compromise, according to a Methodist minister who previously oversaw IRA decommissioning.

Speaking on RTÉ's Six One News, Rev Harold Good said the people of Ireland do not want a quick fix with a short shelf life.

Instead, he said they want political leaders to engage in an "honest and compassionate struggle" to bring together something they can live with.

Rev Good said the political realities in Northern Ireland have been "brought very much to the fore" by Fr Magill's speech at Ms McKee's funeral yesterday.

He said that he spoke with Fr Magill leading up to the funeral, but said the Catholic priest could "never have anticipated that it would have such a response".

"That question was uppermost in the minds of all those in the cathedral," he said.

Rev Good added that Fr Magill had not even finished the question when the congregation anticipated what he was going to say and the applause resonated as the crowd were each asking themselves the same question.