Paul Givan has said if the DUP gets everything it wants with regards to the Northern Ireland protocol they will nominate a deputy First Minister.
Speaking on UTV's View From Stormont he said: "We will appoint a deputy first minister to take on that role."
The party remains very clear that it will not be entering an Executive until the post-Brexit trading arrangements are resolved.
On Monday, the Assembly failed for the second time to elect a new speaker.
The DUP refused to back the bid, maintaining its position that it wants to see action from the UK Government over the Northern Ireland Protocol.
The two nominations for the role, the Ulster Unionists' Mike Nesbitt and the SDLP's Patsy McGlone, failed to secure the necessary cross-community support from MLAs.
The plenary session of the Assembly was then suspended as business cannot be carried out without a speaker.
Reacting to the move, Michelle O'Neill says the DUP is "punishing the public" and vows that party will try again to get a new speaker elected.
"We had our second attempt today to try and get things up and running but unfortunately the DUP stand-off continues," she told media in the Great Hall.
"We will come back again, we will do this again, because I'm not giving up. I believe in making this institution work.
"The DUP are punishing the public. It is the public who are missing out. It is the public who are being held to ransom because of their actions.
"That being said, it is incumbent on all of us who have been given a mandate to try to make politics work. It is incumbent on all of us who want to be around the Executive table, who want to be taking on the challenges of today to support workers and families through the cost-of-living crisis, to keep coming back and try to actually get this up and running.
"The public voted for us to roll up our sleeves and provide solutions to the problems of today. I'm determined to try to do that. I'll keep coming back to this if that is what it takes to get the DUP over the line and work with the rest of the parties."