I don't think that Belichick wants to "figure it out". They could make the new money hit less than $20M plus the $6.75M that is part of the old contract. That would give the patriots approximately $21M of cap room to work with. BTW, the available cap room would be only a bit lower if Bill brought in another vet.
Could Bill and Casserio build a contender with $21M of cap money, plus the restructures and replacements? Sure they COULD. The defense would restructure Hightower and Gilmore, and re-sign McCourty and Shelton and continue to be a top 5 defense. The offense would have draft picks and $20M to work with, plus they could replace Sanu and/or Cannon without any additional cap cost. Andrews and Develin would either come back, or free up cap space.
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I believe the problem is that working it out require a 2 year contract, plus a fake 3rd year. If the CBA is signed in the next month, then a 1-year contract like last year might work (if Tom were at all interested).
I suspect that Tom is already deciding between TENN and Las Vegas, and might listen to to other offers. Sure, the team and Tom's folks will have a meeting or two, to see if either has moved enough to continue talks.
A difficult job for Bill is to get what he can in trade for Edelman, from Tom's new team.