DUP leader Arlene Foster has said there is no revolt in her party after Sinn Féin surged at the polls.
The Democratic Unionists saw their 10-seat lead as the biggest party at Stormont reduce to just one following Thursday’s election.
Northern Ireland Secretary James Brokenshire has been meeting Stormont party leaders in an attempt to persuade them to form a new power-sharing executive.
The parties have three weeks to overcome their differences.
Mrs Foster said she was going into the talks wanting to do a deal.
Sinn Féin has insisted it will not go into government with Arlene Foster as First Minister before there is clarity on her involvement in implementing the botched RHI scheme.
When asked about the possibility of nominating someone else to the post, Mrs Foster appeared to keep her options open.
“It’s a matter which Sinn Féin wants to seem to draw red lines about, we’re not drawing red lines, we want to see the negotiations working for the people of Northern Ireland,” she said.
“We increased our mandate by over 23,000 votes, our vote was up in every single constituency across Northern Ireland and therefore we very clearly speak for unionism now and we go into these negotiations wanting to promote unionism, wanting to promote the union and wanting to see a very solid place for Northern Ireland within the United Kingdom.”
When asked about feelings within her party at a press facility in front of the Lord Carson statue in the Stormont Estate, she said: “There is no revolt.”
Deputy leader Nigel Dodds added: “This is the biggest vote cast for any party in an Assembly election, it would be a perverse thing to say that as a result of that, that someone should step aside.”
Sinn Féin closed the gap on the DUP to a solitary seat while the overall unionist majority at Stormont was lost.
After the party met at Parliament Buildings earlier, northern leader Michelle O’Neill said political unionism "must live up to its responsibility to share power".
"From today we will start meeting with the other parties and the two governments and tell how we see it and what it'll take to fix it," Mrs O'Neill said.
"We will confront the two governments who have shamefully undermined and failed in their duty to protect, nurture and develop the Good Friday Agreement.
"What is required is genuine power sharing and partnership with republicans, based on equality and respect and mutual respect for our Irish national identity."
In a later statement, the DUP said after meeting with a Sinn Féin delegation, the parties agreed to meet again on Tuesday.