But this Linksys router also has a subnet of 192.168.1.x.
I want to solve this problem by connecting (using cable connection instead of wireless connection) both computers to a 2nd router (Linksys E1000).
(EDIT: This sentence has been corrected.) When I use VPN (specifically Cisco Anyconnect Secure Mobile Client) on one computer, that computer can not be accessed by other computers on the 192.168.1.x subnet, but that computer still can access other computers on the 192.168.1.x subnet. Two computers are connected to the ASUS router via wireless connection (subnet 192.168.1.x).
I have an ASUS router at home, which is connected to the modem to access the internet.