If you see how the big clubs operate when they're courting a player, they very publicly make it clear that the player would be a great addition. Former players, current players, and coaches are found quoted in the press all praising the transfer target. You get Coutinho mentioning how much he admires Sanchez's playstyle and uses it to mold his own, and so on.
Then you get your key people directly on the line with the player. Have Rodgers talk to Sanchez, and explain his philosophy and why it will succeed. You reach out to Mascherano, and have him hype Liverpool as a perfect destination for Sanchez in the Spanish papers.
If nothing else though, you stump up more dough. 30k more in wages than Arsenal are offering? Sure, but you have to take the deal now. Oh wait, Gerrard is in the room too, and he's selling you on the prospect of Liverpool. Alexis, do you remember that trophy case you walked by on the way in? How about an extra 5M Barca, if you encourage Alexis to join us?
And if all else fails, we explain to him what a terrible person Piers Morgan is, and how he's Arsenal's biggest supporter.