What they might do is make Asante a "non-exclusive" franchise player, which would mean offering the average of last year's top five at the position, which according to Miguel, speaking through the mouthpiece of Da Bruinz, is $7.225 mil.
Given the amount of cap space that most teams have this year that they did not have last year during free agency (due to negotiations on the bargaining agreement), I think we'll see salaries spike up, so making Samuel an "exclusive" franchise player - i.e. giving him a paycheck of the average of the top 5 corners this year, calculated after some date (?) in April - would probably be significantly more expensive.
This also means that if you were ever going to tag a player with a "non-exclusive" franchise tag, this would be the best bang-for-your-buck year, before player salaries start reflecting the new cap.
The "downside" to making Asante "non-exclusive" is that another team is free to beat the Patriots contract offer, but the Patriots have a right to match, or if they decline they'll receive two first round picks in compensation. Considering how well the Pats draft in the first round, and how cheap five years of a first round pick are versus, say, three years of a talented veteran like Samuel, who here wouldn't like that transaction?
One last thing, based on previous performance, if we were to lose Samuel, I'd pencil in Randall Gay over Chad Scott. Both corners have missed a lot of playing time these past few years, but when Gay has been healthy he's generally been in the starting lineup... Scott, not so much.