[coski]

It is related to the HVAC system. See the threads below for more information. It's unclear if anyone has found a solution that quiets the noise. In my experience, it's most noticeable at cool to cold temperatures, but not extremely cold (like 0 degrees F or below). I suppose that's because the heat pump is only effective down to a certain temperature and below that it stops operating.

https://www.e-tronforum.com/threads/purring-sound-from-front.4710/

Pump sound from front end? - AudiWorld Forums

 

[coski]

Here is a diagram of the HVAC system. Maybe you could start by looking at the various hoses in this area to see if there are any that might rattle against another surface. This page is from the "Heating, air conditioning" workshop manual that you can find at this link:
https://cardiagn.com/2019-audi-e-tron-gen-ge-service-and-repair-manual/

 

[coski]

Here is a Technical Service Bulletin with more information and a possible solution.
https://static.nhtsa.gov/odi/tsbs/2022/MC-10210512-0001.pdf

fixed the link - thanks

 

[coski]

Also note that at least one owner reported that the sound only disappeared after they replaced the compressor, so dealing with the hoses and mounts may not be enough.