Earlier this week former SNP leadership candidate Kate Forbes confirmed she was not standing and backed Mr Swinney, having been promised a “significant” cabinet role if he becomes first minister.
If no other candidate meets the nomination deadline, Mr Swinney would be free to seek parliamentary approval to become first minister.
Mr Yousaf has decided to stay on in the role until a replacement is selected.
Once his resignation has been accepted by the King, parliament has 28 days to select a replacement.
There will then be a vote in the chamber to decide the new first minister, which is passed by a simple majority.
The SNP has 63 seats in the parliament, which means it does not have a majority, but the vote is likely to pass regardless.
The parliament’s presiding officer then recommends to the King that the winner be appointed as the new first minister.
A swearing in ceremony at the Court of Session in Edinburgh could take place as early as Wednesday.
At that point, Mr Swinney would officially become first minister.
