Yes, you can, but the question is will you, if I say that with pre ground beans, you will not get the best tasting coffee? Let us not confuse you, and talk about it comprehensively.
A French press coffee maker can help you to make the best coffee at home that you will not find anywhere. You need to use the right kind of coffee beans and stick to the basic rules to enjoy a refreshing and tasty coffee.
The taste of your coffee largely depends on the type of coffee beans you are using apart from the water temperature, water and coffee maker.
If you want to use the regular pre-ground coffee, it is fine, except your coffee will not be anywhere as good as it could be. Using pre-ground coffee beans for making French press coffee is not recommended because it makes the coffee strong and bitter.
One major issue with pre-ground coffee bean is that no matter how fancy they look, they taste nowhere near as fresh as bean you grind yourself. Even if you keep the coffee beans in an airtight container, it will get a bit stale as compared to home ground or fresh ground beans.
Another issue with pre-ground coffee bean is that you cannot grind them in traditional drip-coffee makers. Traditional coffee makers usually produce small to medium-size grains because almost all drip coffee makers use paper filters, which is not appropriate for French press coffee. For French press coffee, the grinds need to be a bit coarse so that it is less likely to squeeze through the holes in the press. You can try big brands that use larger grinds, but they are not easily available.
If you still want to use pre ground coffee beans only, then the best you can do is to buy pre ground beans from a grocery store and ask to grind the beans there. The grinding machine offers diverse setting options, to produce grains of different sizes.
Let the shopkeeper know you need large grains, as you will be using it in a French press.
Even if you are using a large French press, you can only make three or four normal cups of coffee. So, if you want to make more than that, make stronger coffee and pour more hot water in cups. I recommend pouring it into insulated mugs or carafes otherwise the coffee will get cold fairly quickly.
It is advisable to prefer fresh ground or home ground beans over pre ground coffee beans to make French press coffee. Always keep the beans into airtight container to preserve flavor.