Loud hum from doorbell transformer

I just bought an 1923 home which has a neat working door bell which may be original. Anyway there is a very loud buzz originating from the transformer.

Could it be as simple as swapping out the transformer to eliminate the hum or is there any other trouble shooting I should do first?

When a transformer gets old, the varnish and the rivets can get loose, and the plates will hum. It can also hum or get worse if it is overloaded. But the easiest thing is to replace the xfmr and see where you are. Try to find the output voltage and a volt/amp spec. number on it somewhere, and get one with the same output voltage, and at least the same minimum V/A

If it humms all the time, replace it if it bothers you. If it only humms while it is under load (i.e., the doorbell is being used), it could be too small. Best to look at the doorbell itself. They usually have a VA rating on the label. If your transformer is at least that large, it will work well.