Pressing the case near the fan doesn't necessarily mean the fan, it could mean something else is rattling, and pressing the case alters that. Perhaps because of vibrations from the fan, perhaps not.
But starting with the possibility it's the fan itself:
Check if there is anything close to the fan - cables for instance
With the PC off, check if the fan is loose at any of its corners. Tighten the screws if so, but be careful not to overtighten as some plastic may break if you do.
Disconnect the fan and see if the noise is any different.