Asante is a product of the system. When it works he is a pretty good ball hawk, a capacity he always had except it took a few years of working on his hands to be in position to capitalize on it.
Last year he couldn't have gotten an offer based on his performance on a struggling defense. This year in a spotlight game against the top NFC team whose achilles heel is a young QB beginning to struggle in the second half because he's being encouraged to take shots down the field, and playing with one of the best DL's in the business when they are healthy, he turned in a stellar performance. Caught the talking heads attention and they immediately began touting a guy whose secondary they've been lambasting for the last 18 months. I'm sure his agent is positively giddy.
The way I read Asante he will say mostly the right things about wanting to stay, but he has been focused on this payday for some time now. They have had some preliminary talks based on what he is in Belioli's estimation, and those went nowhere indicating the sides are miles apart. It's shaping up to be a Branch like situation only he couldn't hold out this season because he was so abyssmal for most of last season.
I would guess they will do whatever represents the most value to them. Probably tag and trade if he remains a hot ballhawk down the stretch, or perhaps transition tag him if they think his market has cooled. BB values the position more highly than WR, but there are a couple of positions on defense he truly values more. And we have issues with those short and long term that trump overpaying for a system corner.