This isn't the first time the media has caught wind of Butler's desire to get a new contract, and this won't be the last we hear of it.
According to NESN, the Patriots can tender a contract to Butler and pay him $4 million, which is still an underpayment for a player of his caliber. However, if a team were to sign Butler to a contract and the Pats don't match it, the team that signs him would be forced to give up a first-round pick to do so.
By virtue of that, that may seriously limit the amount of teams that want to go after him.
The Patriots shouldn't have as much of an issue locking up Butler like they might with Dont'a Hightower, but I'm of the opinion that they should pay both of them whatever they want. Butler and Hightower literally saved Super Bowls from going the other way for the Patriots and remain two of their most productive players.