Any contract offer has to be considered vs his 'existing' possible contract
2017 - $3.91MM (RFA)
2018 - Team option for $19MM (franchise tag)
And the Patriots would get to go year to year on that, which significantly mitigates risk.
For them to 'tear that up' and offer him an extension, it would have to make sense for the Patriots, and Butler would have to give something up.
For those feeling bad for Butler, remember it still is $4MM guaranteed, remember that there is a good chance he is not in this position without the Patriots taking a chance on him back in 2014. If that doesn't happen he realistically is not where he is today, considering offers for $10MM/yr.
I think the Patriots are playing this fine - they hold the cards in this situation. As noted, the only one not playing it fine is Butler's agent who seems to be getting exposed as a rookie here.
Hopefully they find a middle ground on an extension, maybe a 3-4 year deal. The Patriots made a shrewd move if you ask me signing Gilmore, increasing their 'hand' in this negotiation. It's these negotiations that have enabled them to be so successful. They take sentiment out of the negotiation, they don't 'backpay' for past performance, and they are willing to let players walk if $$ exceeds their estimated 'worth'. The sentimentality / feeling bad for Butler seems really misplaced here, and probably is due to one play in the Super Bowl a few years ago