Persistence is a key to success. That said, the mere act of dogged pursuit doesn't mean you deserve success, and after a while it shifts to insanity. All of this is to say that Juventus still, evidently, think that they can pry Alexis Sanchez away from Arsenal this summer, according to Tuttosport (here's Daily Cannon's translation).
Supposedly they're offering around £37 million, which probably isn't enough to get a counteroffer considering it's barely higher than we paid for the player two years ago and the transfer market has gone (even more) insane since then. The club and Alexis's agent are allegedly in contact, which makes sense if it's true – an agent's job is to make as much money as possible for his player, so of course he'd entertain all options.
It's not happening this summer, though, barring something like some kind of ultimatum from the player. Alexis still has two years remaining on his contract, and the priority's going to be extending that deal, not selling and replacing him. I'd be seriously surprised if anything comes of this, so for now don't worry too much about it.