As soon as rumors started to swirl about Seattle defensive lineman Red Bryant's impending release, he was rightfully connected to the Patriots. New England was connected to Bryant back in 2012 as the Patriots were still in flux about their defense.
You'll remember 2011 being the first season the Patriots really featured a 4-3 front, as Andre Carter and Mark Anderson emerged dominant players in the Patriots system. However, the Patriots didn't really commit to the 4-3 until they drafted Chandler Jones in the 2012 draft, after their plans to sign Bryant fell through.
Bryant is a mountainous 6'4, 325 lbs, will be 30 this season, and would play LDE in a 3-4 front. He played 5-tech DE in the Seahawks' four man front, allowing him to collapse the pocket from across the offensive tackle.
Bryant's role in the current New England defense is pure speculation. I don't believe he would fit in the 4-3 front as an inside NT, and he doesn't provide enough pass rushing value to play him at 4-3 DT. The Patriots can't/won't use him in the same way as the Seahawks because Chandler Jones and Rob Ninkovich both deserves snaps to be on the field.
Now the Patriots will be connected to Bryant until he is finally signed by a team. Just like they'll be connected with Steelers FA WR Emmanuel Sanders. But I believe at the end of the day, the match is no longer there. The Patriots will kick his tires purely out of due diligence, but don't expect them to be banging the table to bring him into the fold.
Edit: As pointed out by Eric Edholm of MMQB/Shutdown Corner, Mike Lombardi loves him some Red Bryant. He went so far as to call him a blue chip player before the 2012 season. There's no coincidence that was the off-season the Patriots went hard after Bryant. I guarantee that Lombardi was the little birdy in Bill Belichick's ear. We'll see how Lombardi impacts this free agency period for the Patriots.