Yes, of course it can. For instance, consider a loudspeaker, modelled as a rigid piston which can be driven forwards or backwards. Now drive it with some suitable alternating signal, gated by a step function. By adjusting the step function suitably then the initial movement of the piston will be backwards, and therefore the sound it emits will start with a lower pressure than ambient. If you have a good microphone and the frequency is low enough you can see this.
It's also easy to see that you don't somehow need to start with a compression to propagate the signal. As a very simple example of that consider a sealed container of which one wall can be moved. Now, rapidly (but below the speed of sound in the gas in the room, to avoid shockwaves), move that wall outwards. This causes low-pressure in the vicinity of the wall you've just moved, and that low pressure rapidly moves outwards across the room until the system equilibriates. How fast does it move? Well, it moves at the speed of sound, because it is in fact a pulse of sound moving across the room.
You can in fact do something pretty close to this thought experiment with a loudspeaker (see caveats below). You want a fairly large speaker (12-18 inch cone, say), and you want to install it in a sealed cabinet, or alternatively in the wall of a room so that the back of the speaker is isolated from the front. Then feed it a suitable step function so that the cone jumps backwards (you can verify the direction the thing moves in by just looking at the cone). This will make a fairly significant thump. When you turn whatever was generating the step function off you'll get another thump as the speaker relaxes back to its neutral position, and again you can see the cone move.
Notes on the above experiment. (1) You don't in fact need the back of the speaker to be completely isolated from the front, what you want is that the path from the back to the front is long enough so that the corresponding step pulse from the back of the speaker does not propagate to the front too quickly. So in particular you want it either in a sealed cabinet or in the wall of the room, what you don't want is a 'ported' cabinet. (2) You can do this with small speakers but it's harder to see and much less impressive. (3) Be careful when doing this as speakers often have pretty low DC resistance and applying DC to them is a good way to fry them as the voice coil will get very hot.
I have, initially inadvertently, done this experiment by playing with an amplifier which had a significant DC offset and some walled-in studio monitors. I was probably lucky I didn't destroy them, especially as I was a guest of the owners at the time.