You can certainly have an honest discussion your manager and volunteer. Your traction will depend on what kind of manager you have, the type of relationship you have with them, and their relationships with their own management chain. If conditions and timing are right, it is possible to volunteer and receive a severance. In my experience, most low to mid-level managers absolutely do not want to layoff their staff as their teams are already stretched thin, but are forced by higher ups to cut certain number of positions and then deliver the terrible news to employees. If you can take the place of someone who is unprepared to lose their job or otherwise wants to stay, it makes sense for the manager to put your name on 'the list'. You're happy because you get a one-time payout and your manager is happy because they don't have to crush an unsuspecting employee and their family.
Of course this could totally backfire on you as others have indicated (why pay you severance if you're likely going to leave on your own). You'll have to consider the circumstances of your own situation before starting down this road.