Where does the magnet stick to?
The "brass" or the payload?
Either way, if it's the payload that does have steel in it, the barrel will be fine because the shot cup/wad is actually what's holding the bulk of the pellets as it's fired through the barrel. Mind you, not all of the pellets will be securely in the shot cup, but most of them will be.
The "brass" or whatever material that Winchester is using these days, shouldn't hurt your chamber either.
Edit, I just checked some Remington hulls that I've reloaded a half a dozen times, the "brass" on them is slightly magnetic too.
Won't hurt anything.