Check whatever router(s) you are interested in to be sure. Read the specifications carefully. Check manufacturer FAQs and forums.
My Linksys 1900ac has two USB ports; one 2.0 and the other 3.0.
I use the 3.0 port with a flash drive as a destination for downloads. Has been sensitive to some USB flash drives from time to time. And I do have a 3.0 USB external hard drive with its own power supply. That drive is used and moved about between PC's for backups, miscellaneous testing, and use. Very flexible.
I do not like to place "heavy" loads such as HDD's or DVD/CD players that depend on USB ports for power. Not so bad for a desktop maybe. However not what I would do on a router.
Smorizio's idea is more the sort of thing I would do. Put the HDD in an external case with its own power supply and ethernet port. Plug the "NAS" into a router LAN port.