It comes after North Belfast MLA Gerry Kelly said the relationship between the DUP leader and his counterpart Martin McGuinness has broken down.
Meanwhile Sinn Féin President Gerry Adams has accused Mr Robinson of a failure in leadership.
Speaking to UTV, the First Minister denied the party's claims and said the Stormont administration is not in any risk.
"I've been involved in a number of crises over the years and I have to say our present circumstances don't have that feel about them at all," Mr Robinson said.
I don't get any impression that anyone has any intention to bring down the administrations.
"Of course there are big problems, there always have been, and it's important that we address those issues, but one man's crisis is another man's problem to solve."
Deputy First Minister Martin McGuinness has also spoken to UTV and said he is in "problem solving mode" after holding several discussions with Mr Robinson in recent days.
He continued: "I've had several lengthy conversations with the First Minister yesterday and today and it's with a view to finding a solution to the problems we face at this time.
"There have been different views on how the course of the past year has been dealt with and people have expressed various opinions on what is happening within unionism.
"I'm not going to go into that in the context of this interview except to say we've had very serious discussions about some very serious problems."
Relations between Stormont's two largest parties have been tense since the DUP withdrew its support for controversial plans to build a Peace Centre at the former site of the Maze.
The move followed a summer of violence surrounding parades and the unrest over changes to the flag flying policy at Belfast City Hall.
I'm in problem solving mode.
Mr Robinson's shift in position on the Maze was linked to the staging of a republican commemoration in Castlederg, Co Tyrone, where Gerry Kelly gave a speech which unionists said glorified terrorism by remembering two IRA men who were killed by their own bomb.
Earlier, Gerry Kelly claimed the crisis at Stormont "emanates from a crisis in unionism".
He continued: "Of course there's a crisis - it's seen on the streets, it's seen through the months of sectarian violence, and we've had the First Minister missing.
"When you look at Martin McGuinness regarding the dissidents, he is first out saying what the situation is and defending the institutions, partnerships and the way forward - but when it's Peter Robinson he is nowhere to be seen.
"This can be fixed but it needs two people in partnership."
Former US diplomat Richard Haass has been tasked with leading all-party talks on issues such as parades, emblems and how to deal with the legacy of the conflict.
He is due to return to Northern Ireland next month, but asked the parties to continue to meet while he's away - after leaving with the message that there is a "strong sense of possibility" that the December deadline for consensus between politicians can be met.
Ulster Unionist leader Mike Nesbitt said he sees no crisis at Stormont.
He continued: "There is just some competitive posturing between Sinn Féin and the DUP. But there is a fundamental issue which has raised itself to the surface again and it is, who really wants to share power, share space and share a future.
"The Ulster Unionist Party will not be deterred by the current spat. It is time to deliver and do what's right for Northern Ireland."
Alex Attwood of the SDLP said: "Sinn Féin talk crisis and hope people will jump.
"There are certainly very big problems, created by the DUP and Sinn Féin, but the way forward is through Haass and a return to the hopes and ambitions of the Good Friday Agreement."