Under French law, an owner is free to sell to whomever he or she wants. Therefore, in principle, nobody can force you to sell your house to them. There is one exception, namely the situation of a "public offer". A "public offer" exists when you have published an announcement stating your clear intention to sell the house, and the announcement contained sufficient terms for a contract to be concluded, in this case the identity of the house and the fixed asking price. In this case, the first person who comes up with the money is entitled to have the contract concluded with them, unless very exceptional circumstances apply. For example, you could state that you are not selling the house anymore, but then you might have to pay damages for the loss of time and any expenses of the willing buyer. Or you could find some imperative need to keep the house, which appeared after you made the announcement. Otherwise, they can sue for a judgment in lieu of contract.
If you did not make a public offer on those terms, then you can always change the price, or say that you changed your mind.
Be careful whom you refuse. There is also the possibility of being sued for discrimination, if the willing buyer is a member of a religious or ethnic minority, has an infirmity, or is otherwise able to claim that you discriminated for some reason, by refusing to sell to them. This may lead to higher damages and / or fines.
I hope my answer is useful. Even though it might not be a very positive news, I believe it is important that you know where you stand. I look forward to your rating, which is essential to my activity.
Dr I L Vlad