Butler signed his restricted free agent tender on Tuesday, but is still waiting on a potential trade.
However, New England is still open to trading him, according to Patriots director of player personnel Nick Caserio.
If Butler stays on with the Pats for the 2017 campaign, the first-round tender would earn him $3.91 million, which would be a major step up from his $600,000 base salary last season.
"We had a chance to visit with him", Payton said of Butler at the time.
With it becoming clear no team would sign Butler to an offer sheet - the deadline is April 21 - he chose to sign the contract, which now gives the Patriots an opportunity to discuss a trade with other teams.
On whether Garoppolo is "not for sale": "I'd say to answer your question, anything that we do, we're always going to do what we think is in the best interest of our football team and that's how we approach it". As the draft gets closer it will be worth watching Butler as the Patriots have no picks in the first two rounds and Butler could be on another team to start the season.
With the draft just over a week away, there could be more rumors between now and then, but it appears Garoppolo will be spending another season backing up Tom Brady in New England. He was named a Pro Bowler in 2015 and has helped win two of the last three Super Bowl titles, famously grabbing the game-sealing interception in Super Bowl XLIX.