The owners of the Argyll farm, Mowi, confirmed that almost 50,000 salmon escaped after the farm was damaged during Storm Ellen.
Fisheries Management Scotland (FMS) said “significant numbers” of farmed fish had been caught.
The salmon, that can be identified by damage to their fins, could pose a threat to wild stock.
Farmed fish have been reported in rivers across Loch Lomond, Ayrshire, Clyde and Argyll, and FMS have warned they could also show up further afield.
Its aquaculture interactions manager Polly Burns said the salmon could be distinguished from wild stock by the damage to their fins.
She added: “Given the risks these fish pose to the genetic integrity of Scotland’s wild salmon populations, we appreciate the ongoing vigilance among the angling community. It is essential that this continues and that anglers are clear about what to do should they capture such fish.
“We also need to understand any impacts this escape will have on our wild salmon through interbreeding of wild and farmed fish. Fisheries Management Scotland are working with Marine Scotland Science and Mowi to develop a genetic monitoring programme.”
The fish escaped from the farm in North Carradale farm, near Campbeltown, when sea bed anchors on four of the 10 fish pens became detached during Storm Ellen.
Anyone who catches an escaped fish has been asked to humanely kill it and take a sample of its scale for confirmation that it was farmed.
FMS has produced guidance on how to identify farmed fish and the steps that should be followed and asked captures to be recorded in its central reporting form.