Sinn Fein Stormont leader Michelle O’Neill spoke with the new Northern Ireland Secretary by phone following his appointment.
New Northern Ireland Secretary Shailesh Vara has said that his “immediate priority” will be to restore the Assembly and the Executive as soon as possible.
Earlier, Mr Vara was urged to work constructively and even-handedly to restore power sharing at Stormont.
The Assembly has been unable to function for several months after the DUP withdrew from the powersharing administration.
Sir Jeffrey Donaldson has said his party will not nominate ministers until the UK government takes action against the Northern Ireland Protocol.
Mr Vara was appointed to the role following the resignation of Brandon Lewis early on Thursday.
Mr Lewis was among a revolt by Conservative ministers against Prime Minister Boris Johnson, who later announced his resignation.
Mr Johnson is set to remain as Prime Minister until a successor is in place, expected to be by the time of the Conservative Party conference in October.
Mr Vara spoke with the leaders of the Stormont parties by phone on Thursday afternoon.
In a statement released on Thursday, Mr Vara said he was “very pleased” to speak with Sinn Fein Stormont leader Michelle O’Neill, Sir Jeffrey Donaldson, UUP leader Doug Beattie and SDLP leader Colum Eastwood following his appointment.
He said he was looking forward to speaking with Alliance Party leader Naomi Long and engaging with others “in the coming days and weeks ahead”.
“My immediate priority, and that of the Government, remains restoring the Northern Ireland Assembly and the Executive as soon as possible,” he said.
“By working together we can create a more prosperous Northern Ireland with a strengthened and growing economy, as well as reconciling issues of the past and taking decisive action on the issues that matter most.
“I look forward to playing my part, along with my ministerial team, to make Northern Ireland a better place to live, to work and to invest, and to strengthen its place within the United Kingdom.”
Ms O’Neill said she told Mr Vara during their conversation that he has a “responsibility to work constructively and even-handedly to restore powersharing”.
“I told him that he is the seventh British Secretary to fill the post under this Tory Government, and that while their priorities lie elsewhere, it is his duty for however short his tenure, to work constructively and even-handedly to restore powersharing,” she said.
Ms O’Neill said she also told Mr Vara that she felt his predecessor had “undermined political stability at every turn”.
“This included giving cover to those denying the public democratic representation at the Assembly they voted for, and refusing to respect the democratic outcome of the recent election,” she said.
“The British government needs to accept its responsibilities and start working with all the Assembly parties, and the Irish Government to restore the Good Friday Agreement institutions, and stop placating the DUP who are in a needless stand-off which is unjustly punishing the public.”
On Thursday evening, Ireland’s minister for foreign affairs Simon Coveney had “a warm and friendly phone call” with Mr Vara, according to a statement.
“They agreed that Northern Ireland needs a Stormont government up and running as soon as possible. They vowed to work closely together on this and many other things and to meet in person as soon as possible.”
Mr Coveney previously worked with Mr Vara when he served as Minister of State at the Northern Ireland Office in 2018.
Earlier Mr Vara said it was a “huge privilege to return to the Northern Ireland Office as Secretary of State”.
“Having served previously in the department I recognise the immense contribution of Northern Ireland, and its people, to the UK and I look forward to representing those interests at the Cabinet table,” he said.
Mr Lewis announced his resignation early on Thursday, tweeting a copy of his resignation letter in which he told the Prime Minister that he had “given you, and those around you, the benefit of the doubt”.
“We are, however, now past the point of no return.
“I cannot sacrifice my personal integrity to defend things as they stand now. It is clear that our Party, parliamentary colleagues, volunteers and the whole country, deserve better,” he said.
NIO minister Conor Burns indicated he will remain in post, tweeting that he will lead for the government at a meeting of the British-Irish Council.
“It is vital at this time that the job @NIOgov continues to be done.
“I am committed to continuing to serve the wonderful people of Northern Ireland to the best of my ability at this tough time,” he said.
The Stormont Assembly remains collapsed amid a row over the Northern Ireland Protocol.
The DUP is refusing to nominate ministers to the executive or a speaker to allow the Assembly to function until the UK government takes action on the post-Brexit arrangements.
Ms O’Neill was also critical of Mr Lewis over his proposed bill for dealing with Northern Ireland’s troubled past.
The Northern Ireland Troubles (Legacy and Reconciliation) Bill offers an effective amnesty for Troubles crimes for those who co-operate with an information body.
She added: “He has undermined the Good Friday Agreement and political stability, and his actions gave us a daily reminder why those with no mandate or votes here will only ever act in their own political interests, and not ours.
“The political stability of the North cannot be a hostage to the Tory-in fighting, Westminster chaos and continued DUP disruption.”
Prominent victims campaigner Raymond McCord welcomed the resignation of Mr Lewis.
Mr McCord said: “He (Mr Lewis) has betrayed the victims of the Troubles with the Legacy Bill that gives an amnesty to murderers.
“Brandon Lewis decided to reward the murderers with amnesties for the murders they committed, and punish the victims and their families by ensuring all routes to justice and even civil cases are closed down.
“He insulted all victims by claiming it would help us.
“He certainly will not be missed by victims.”