Butler visited the New Orleans Saints last month, but they declined to sign him to an offer sheet that could have required them to compensate New England with their first-round draft pick, No. 11 overall. Instead, the Saints are believed to covet Butler in exchange for the No. 32 overall pick they received in the trade that sent Brandin Cooks to New England.
Butler could not be traded until he signed the tender, which would pay him $3.91 million for the 2017 season.
Patriots director of player personnel Nick Caserio was asked several times about Butler at his pre-draft news conference on Tuesday, but declined to tip his hand.
''Right now we're focused on getting ready for the draft,'' he said. ''That's where the focus is. I don't anticipate that many transactions here between now and next Thursday or Friday. Could that change? I mean, look, it's the National Football League. Things can change in a heartbeat.''
Butler, now 27, was an undrafted free agent who during his rookie season of 2014 intercepted Seattle Seahawks quarterback Russell Wilson at the goal line in the final minutes of the Super Bowl to secure the victory for the Patriots. But New England coach Bill Belichick has balked at giving him the lucrative, long-term security he has been seeking.
Instead, the Patriots gave the money to free agent Bills cornerback Stephon Gilmore, a 2012 first-round draft pick who made the Pro Bowl last season after recording 48 tackles and a career-high five interceptions. They also traded a sixth-round draft pick to Indianapolis for tight end Dwayne Allen and a second-rounder to Carolina for pass rusher Kony Early.
Also Tuesday, the Patriots signed tight end Matt Lengel and released offensive lineman Tre' Jackson.
For more NFL coverage: http://www.pro32.ap.org and https://www.twitter.com/AP-NFL