It isn't avast.svc there is no such file, but avastSvc.exe this is the main scanning engine and controls all the shields. Buy if you have none of the real-time shields enabled I haven't a clue what is going on.
So is it the avastSvc.exe file ?
I have no idea why it would need to be connecting to avast.com, but you don't give is anything to work with, ports, protocols, IPs if possible as there are many different IPs for avast.com including all of the update servers, but that would normally be the avast.setup file connecting to those.