I hate to say this but I really don't think this is going to go down until close to the end of the close season, maybe even the entire window, if it does (I think it will).
I know this will be maddening to a lot of our fans, but Arsenal will surely be in no rush to sell, especially to us. Maybe Wenger hopes Alexis will like playing with Lacazette and stick around.
Txiki and Pep surely know we can survive without him, having SJS available during preseason, who are already familiar with the way we play and provide us with 2 of the 3 to start the season if we have to. This is also why I also think the high priority (pretty much needs to happen) is to get 1 right sided and 1 left sided fullback signed by July 10th or so (seems to be Alves and Mendy) because we can add the other two after and they will just get less games to start the season as they blend in.
Jesus would seem to be the favourite for that place, if I was wagering anything, and then Sane and Sterling...in that order of probability. But that's also part of why we have preseason, no? To let players prove themselves.
Not signing Alexis before preseason would allow one of them to show who should start as the 3rd attacking player (which would not exactly be shocking for Pep to use as motivation) assuming we sign Alexis. If not Sanchez, nor someone of his quality like say, Coman, the three could prove themselves as one of the two to play with Kun.