My Sony MDR-1000X headphones can be hardwired, as well as wireless(bluetooth). They sound very good, but I'm not sure I'd use them professionally to monitor audio critically. Also, they have to be turned on and still use battery power when hardwired(even when not using the noise cancel function).
Heard these at CanJam in LA- excellent sounding headphones and super comfortable: https://en-us.sennheiser.com/hd1-wireless-headphones-with-mic . Can also use with your phone (has a mic).
(for reference, I use custom modded Stax (electrostatics) in my office for critical listening)
My Bose around-ear headphones work via bluetooth or wired. No battery power used when wired. I have used them for monitoring but I'm not doing super critical audio work.
I use them mostly for music listening but they work fine when doing some corporate video work.