I plan to install OpenWrt to a X86 computer which has a built-in Ethernet port and an installed fax modem PCI card. My internet connection is provided by a RJ11 cable. Will OpenWrt connect to the Internet via the fax modem PCI card and render data to the built-in Ethernet port like any digital modem do? I am asking this question because fax modem cards are primarily designed to transmit and receive documents. If OpenWrt does not support fax modem PCI cards, how could I connect the computer to the RJ11 cable? Should I install a softmodem PCI card in the computer?
Most of those cards are so-called "winmodem." Drivers do not exist for Linux.
If your modem can be used with a new desktop Linux then there should be a way to make it work with OpenWrt. Otherwise you're going to need to replace it. External modems that connect by serial cable will all work. Some USB modems are compatible such as this one: https://www.amazon.com/gp/product/B01MYLE06I/
Make sure it is listed as "controller based" or "hardware based" and Linux compatible. There is another type which typically sells for less than $20 but requires proprietary Windows drivers.
If you're connected by landline, ADSL is probably available.
Thank you, MK24, for the response. I am obliged to you.