From the Instrument Procedures Handbook, page 4-41 (as part of the Missed Approach section), there is a quote about what to do when you need to go around when operating beyond the missed approach point:
In the event a balked (rejected) landing occurs at a position other
than the published missed approach point, the pilot should contact ATC
as soon as possible to obtain an amended clearance.
I have seen several other references in FAA documents which say to "contact ATC as soon as practical" in similar situations, but nothing which specifically says to do it any time that a go around is initiated by the pilot (i.e. under VFR).
That being said, if you have received a specific clearance (for a touch and go), and decide that you prefer to do something else (low approach/go-around), you should always receive an amended clearance first. If you know ahead of time, your best option is probably to request the "option", in which case you have a clearance to do either one.
On the other hand, if the reason for the go-around is due to a safety issue then execute the go-around immediately when needed, and follow the above guidance to contact ATC as soon as practical if they don't call you first.
I was taught by my flight instructor to always report it, and the tower certainly needs to know about it in order to provide proper separation with other aircraft operating in the area. If nothing else, it is a "best practice" to ensure that the person in the tower doesn't get distracted and not notice that you didn't land.