Nest thermostat - electrical buzzing/humming noise

I have recently installed two nest thermostats (the black gen 3, and white model E). The system is gas radiator heat wired to the model E via the usual Rh and W wires, and Rh, Rc, Y, G, W wired to the other one to control both heat and AC. I have no C wire.

Both thermostats emit a quiet electrical buzz, but it is definitely noticeable. This is happening whether heat is on or off. It stops if I take off the thermostat cover.

Any idea what could be causing this? Is this normal? Anything I could check for?

Thank you