The Social Democrats have opted to hold an internal consultation on whether they should greenlight a similar vote.
Their decision is also due on Sunday, and the Social Democrats majority would set the parliament up to officially initiate the application the following week.
Swedish media outlets expect the final consequential vote by Monday, May 16.
Lawmakers should send the formal application to NATO soon after, although there is no expected date yet.
Both countries are riding on widespread public favour and have not felt it necessary to hold referendums on joining the organisation.
Public attitudes towards NATO have reversed since Putin started invading Ukraine, with national polls showing the majority in favour.
Approximately 75 percent of Finnish nationals want to join, while a slimmer majority of 60 percent want Sweden to do the same.