QuincyAbeyie don't agree with how you've framed it. Yes, he moved to Real Madrid, which chronologically made them his first choice but by the time Arteta had brought him on loan for a season and Madrid were ready to sell, we (and Arteta) were very much his first choice. So he hadn't at any point rejected Arteta's Arsenal, which for me is the critical difference.
Arteta's track record has suggested he only wants players who are fully bought into what we're trying to do as a club and if for whatever reason Williams would still prefer Barca over us, I think you could legitimately challenge whether his motivations align with ours.