"The Patriots have a need at the position after LeGarrette Blount signed with the Steelers, but there is doubt that they'd be aggressive suitors for Johnson, in part because of his limitations in pass protection and how highly the team values that skill with its backs. Just don't see the Patriots getting into any type of bidding competition at a position they have shied away from spending even moderate dollars the last two years."
We know Bill Belichick and Johnson seemed to hit it off at the 2011 Pro Bowl. Others, such as former Buccaneers general manager Mark Dominik, have made the connection between Johnson and New England.
In the end, I wouldn't be surprised if the Patriots explore the possibility in the event a strong financial market doesn't develop. They were in on Maurice Jones-Drew (according to the player himself) before he signed in Oakland and had Michael Bush in for a free-agent visit as well.
So we'd expect the team to add a running back, but I have doubts it ultimately would be Johnson.