Yes you can. I own a Xbox 360 S 4GB myself and got my hands on a dead old Arcade 360 with a 20GB harddrive a few years ago.
I opened up the old HDD case and plugged the SATA HDD into the Xbox slim's harddrive bay without any case.
Everything is recognized , since it's a Microsoft signed HDD.
But since it was lose I decided to order an aftermarket housing for the drive.
Eventually the HDD Slim case came after a few weeks from china for a few bucks and I took the drive and put it into the case. Plugged it into the Slim console and it did fit perfectly.
Works like it should do.
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