Just so we're clear (please correct any mistakes):
Butler's a restricted free agent because he has 3 accrued seasons (basically, seasons with 6+ games played).
The Pats had the option to tender him an qualifying offer at one of 4 levels (first round, 2nd round, drafted round, or no compensation); the first 2 are much more expensive but guarantee a compensation pick of that level if he signs an offer sheet elsewhere and leaves the pats. They tendered him a first round qualifying offer, which this year is $3.91 million dollars.
Butler now has the following options:
1. Negotiate a contract with the Patriots (usually this would be a multi-year extension, but it's whatever the two parties agree to).
2. Sign the tender/qualifying offer, which amounts to a 1 year contract at $3.91 million.
3. Sign an offer sheet with another team: he
can do this without signing/accepting the tender. If he does this, New England can either:
a) Match that offer sheet themselves, in which case it becomes the new contract between New England and Butler
b) Decline to match it, in which case it becomes the new contract between the other team and Butler AND New England receives a first round compensation pick from the other team.
4. Not play in the NFL.
Until Butler is under contract with the Patriots (which would happen under (1), (2), or (3)(a)), he cannot be traded, though he could negotiate a trade-and-sign a la Welker with the Pats and another team.
The last date for him to sign an offer sheet with another team is April 21st; after that, he's limited to signing the QO, negotiating a contract with the Pats, or holding out.
After June 15th, the Pats can reduce the QO to $660,000 (110% of his 2016 salary).