The fact Perez has sent his dogs (Marca) loose gives me some hope that they aren't any closer to getting him than they were 3 months ago, seen this happen with Salah around his last contract extension and the Mbappe saga more recently before he re-signed with PSG. This is Perez we're talkin about though, he unsettles players and frankly with signing him on a free, even if he turns out to be a disaster for them or snaps his ACL on day 1 of training, he is in a position here to severely affect the harmony of the squad he knows are their UCL contenders this season. That alone would be worth the signing bonus and wages he will put on the table, not to mention, a bit of a quiet distaste for each other has been present over the last decade or so, it's a major PR win for him and a fat L for FSG if he manages to take one of our own for absolutely nothing in his prime, with his prime being world class.
If he's on the fence, they will put the pressure on with an ultimatum in this window, if he's leaning towards them already, the next round of negotiations with the club will be the last, if he/his agent are playing the contract game then we're going to hear nothing until it's signed. But i firmly believe, given how close he is with Jude, that Perez will know his state of mind and Jude will absolutely be acting as a mediator because he wants his mate with him, and due to that, if there is a chance he's undecided but more likely to stay, they will throw £20m at the problem in this window, knowing it'll be rejected, just to get the circus on the road
EDIT: Oh would you look at that, they've tried the good ol offer in Jan tactic. So predictable