OS Monitor lists network connections by app:
OS Monitor and Connection Tracker listing connections (source: Google Play; click image to enlarge)
As you can see, OS Monitor lists up all connections, and lists the corresponding app "owning" this connection along. This should enable you to see all servers your suspicious app connects to. There are other apps available with the same feature as well, as e.g. Connection Tracker shown in the second screenshot.
If you really need the full URL, the above won't bring you there (both apps use methods similar to
netstat, which only reveals the hosts). Instead, you might want to take a look at e.g. tPacketCapture, which "sniffs" your network traffic (without requiring root), and saves the protocol in so called
.pcap files. These again can be read then by e.g. Shark Reader.