First Minister Arlene Foster has written to her partner-in-government Michelle O'Neill asking her to stand down pending police and Assembly standards investigations into scenes at the funeral of Bobby Storey.
However, Mrs Foster has said her party will not collapse the Stormont institutions if the deputy First Minister refuses to step aside.
The news comes after Sinn Fein announced that Ms O'Neill will not be stepping aside "under any circumstances" amid claims she broke social distancing rules at the funeral.
DUP leader Mrs Foster said she would not share an Executive platform with the deputy First Minister until the funeral row is resolved.
"Why would I punish the people of Northern Ireland for the bad behaviour of Sinn Fein?" she told UTV.
"People expect me to lead, I will lead and I am not going to walk away from the government of Northern Ireland at a time when we are having severe challenges ahead of us. I think that that would be a dereliction of my leadership and I'm not prepared to do that."
On Thursday morning, SDLP leader Colum Eastwood joined the Alliance Party, the Greens and the Ulster Unionists in requesting Ms O'Neill step aside over the issue.
The leaders of the five main Stormont parties are due to meet on Friday to attempt to defuse the escalating row.
In a statement released on Thursday afternoon Sinn Fein remained defiant on the issue.
"Michelle O'Neill will not be stepping aside as deputy First Minister under any circumstances," a party spokesperson said.
Here's how Thursday unfolded: