Butler could not be traded until he signed the $3.91 million tender for the 2017 season. He visited with just one team, the Saints, during free agency.
The Saints never planned to sign Butler but they appear to be interested in a possible trade for the 27-year-old cornerback.
Saints coach Sean Payton said at the NFL meetings last month the team plans to add two cornerbacks this offseason and showed his interest in Butler.
"We had a chance to visit with him,'' Payton said of Butler at the time. "Currently, to my understanding, he hasn't signed his tender and so ... we can sign him to an offer sheet, but I don't think we're going to do that and give up the 11th pick. In fact, I know we're not going to do that.
"It's ongoing, if you will. For us, it was a chance to meet him, put him on the board, find out how much football he knows. I think it was a good step."
Butler has played in 43 regular-season games with 33 starts, including starts in all 16 games over the last two seasons for the Patriots. He has six interceptions and 36 passes defensed.
Butler has played in eight postseason games with five starts, and he had an interception at the goal-line that secured the win over the Seattle Seahawks in Super Bowl XLIX. He was named to the Pro Bowl following the 2015 season.