A fresh round of talks to restore devolution is to begin at Stormont on Monday, with Secretary of State Julian Smith due to hold bilateral meetings with Northern Ireland's party leaders.
The DUP and Sinn Fein are under immense pressure to reach a compromise over an Irish Language Act following poor results in last week's Westminster poll for both parties.
Secretary of State Julian Smith has warned that an Assembly election will be triggered if no agreement is secured by January 13.
Tanaiste Simon Coveney, who will be in Belfast on Monday, said that it was an "important week for Northern Ireland" and added that "with generosity on all sides we can succeed this time".
There will be round-table talks involving the two governments and all the parties later this week.
Sinn Fein and DUP Assembly seats would be at risk if Northern Ireland went to the polls in February or March if last week's Westminster election results were repeated.
On Sunday night, both parties said they would approach the talks positively.
The Secretary of State spoke to all five party leaders on Sunday and later tweeted: "Good calls with all five party leaders this morning. Look forward to starting positive process to get Stormont back up and running."
Good calls with all five party leaders this morning. Look forward to starting positive process tomorrow to get Stormont back up and running pic.twitter.com/O2qPXcfoqE— Julian Smith MP (@JulianSmithUK) December 15, 2019
DUP leader Arlene Foster said it was time to restore devolution.
"People want decisions made about welfare, hospitals and schools.
"That was the clear message of the election. Northern Ireland has been deprived of local ministerial-led government for three years. Central to the talks must be the sustainability of the institutions so never again can one party hold the rest of Northern Ireland to ransom."
"Sinn Fein has barred everyone from government for three years despite other parties willing to take their seats."
But Mrs Foster added: "We live in a divided society and to move forward will require all the parties to step up to the plate. For my part, we will not be found wanting."
Sinn Fein vice-president Michelle O'Neill said the Stormont impasse could be broken if the "political will" existed.
"We need a new kind of politics, a new assembly and a new executive," she said. "We need to deliver good government and properly resourced public services to all. We need an executive which is transparent, accountable and inclusive."
Ms O'Neill said a "credible restored executive" must deliver on "public sector pay, safe staffing levels in the health service, economic policies that deliver prosperity and invest in rural communities, and an appropriate welfare mitigation package".
She added: "We will work towards securing agreement on outstanding issues, including an Irish Language Act, reform of the petition of concern, the legacy of the past and integrity in government."
SDLP leader Colum Eastwood said: "We are entering last-chance talks this week. The SDLP has made bold proposals that are uncomfortable for us but are in the interests of restoring government to meet the needs of patients, pupils and healthcare workers.
"We will continue to stretch ourselves but it is for the DUP and Sinn Fein to set aside their dispute and act, at last, in the interests of the people we all represent."
Alliance leader Naomi Long said an Assembly election must be called if the talks fail.
"Every day this impasse continues, the public is suffering - people are losing their jobs, budgets are being slashed, organisations are going to the wall," she said.
"The health service is currently facing a major crisis but that is likely to be followed in the near future by other crises in areas such as education and infrastructure, It's clear that cannot go on."
UUP leader Steve Aiken said that if power-sharing was restored, the new institutions must offer "sustainable, accountable and transparent" government.
"There is little point in rushing back into Stormont simply because the DUP and Sinn Fein have decided that now the election is over, they are willing to set their own differences aside," he said.
"The operation of the Executive, accountability mechanisms for ministers and their special advisers, as well as the petition of concern must all be subject to major reform."