I'm not even totally sure that offer is a lowball. I think a lot of teams will look at Hightower's role in the Pats' defense and will question if that's something they can reliably replicate. Hightower's one elite skill is run-stopping, and run-stopping LBs don't exactly command a premium. Otherwise, his main value is in his versatility, that he can generate interior rush as a LB, get on the line and rush from the edge, can cover credibly, etc. That's what makes him a three-down LB. When **** LeBeau looks at that skillset, I'm not sure he sees a guy who will produce in his system in a way that justifies more than $10M per year, because LeBeau probably has a pretty good idea of what he wants his LBs to do, and probably values Hightower to the extent that Hightower can fit into his existing system (which he can, you're just leaving some of his versatility, and therefore some of his value, on the shelf).
All along, I figured Hightower would stay here because, of all the Pats' defensive stars that were hitting FA this year, he's the one for him his value to the Pats most exceeded his general value to any other team. He's too unique, and not really an archetypal player at his position, which I figured would likely depress his value a bit to everyone on the outside, creating a situation where the Pats' perception of what he was worth to them would be significantly higher than anyone else's. Of course, I also thought he would've been extended before this season and it never would've got to this point, so I've been wrong at pretty much every step of the way.