I have the same hardware and the same problem. In order to support WOL, the Asus H87I-Plus requires an option set in the BIOS (Onboard>APM>Power on by PCIE). When I have this enabled, I can indeed start the machine using the wol magic packet, but there is an undesired side-effect: when I shutdown the machine, it stays shut off for about 5 sec and then turns back on all by itself. If I disable the option in the BIOS, this does not happen... but I am unable to use wol.
Can you post the output of ethtool eth0 for me? Here is the output of my command:
% sudo ethtool eno1
Settings for eno1:
Supported ports: [ TP ]
Supported link modes: 10baseT/Half 10baseT/Full
100baseT/Half 100baseT/Full
1000baseT/Full
Supported pause frame use: No
Supports auto-negotiation: Yes
Advertised link modes: 10baseT/Half 10baseT/Full
100baseT/Half 100baseT/Full
1000baseT/Full
Advertised pause frame use: No
Advertised auto-negotiation: Yes
Speed: 1000Mb/s
Duplex: Full
Port: Twisted Pair
PHYAD: 2
Transceiver: internal
Auto-negotiation: on
MDI-X: on (auto)
Supports Wake-on: pumbg
Wake-on: g
Current message level: 0x00000007 (7)
drv probe link
Link detected: yes