They may work but only at 11mbs (maximum, in reality always less) because the pci adapter is a wireless B spec.
Judging by the model number the router is wireless G which probably has a theoretical maximum of 54mbs.
You may find an XP driver for the adapter on the Linksys site.
But it might be worth investing in a newer adapter.
There are sometimes issues between old wireless B adapters and newer routers (even when wireless B or mixed mode is selected in the router).