Of course they do. It's a golden ticket to future peace for their nation.

Unfortunately it's not something that has a process. There's no pre-NATO pinky swear stages. Until the war is over they only get politician's promises, which sucks for them and the NATO states.

If the war ended today, Ukraine would be put into the fast track for NATO membership. I don't know if they'd make it (looking at Turkey and other states in the area), but they'd get a hearing.

They've got the coherency and the military chops to be e member state of both the EU and NATO, just not the secured borders to start the process right now.