Butler signed a one-year, $3.91 million tender with the Patriots last Wednesday, which means New England has the opportunity to trade him.
Butler, who was a restricted free agent, had previously not signed the tender in hopes that another NFL team might sign him to an offer sheet before last Friday's deadline. He visited one team, the New Orleans Saints, earlier in free agency.
The Saints didn't plan to sign Butler, 27, to an offer sheet, but at the NFL's annual meetings in March, coach Sean Payton seemed to hint that the club could be interested in a different scenario.
If Butler had signed an offer sheet with the Saints and the Patriots didn't match, the Saints would have had to surrender their first-round pick in the 2017 draft (11th overall) to New England.
The Saints have three picks in the top 42 selections of the 2017 draft and likely can find a cornerback to fill their need with one of those picks.
The MMQB first reported that Butler is likely staying with the Patriots.
The Patriots and Butler's representatives had discussed a contract extension last offseason, and some involved in the negotiation believed a deal was close to being struck. But things were never finalized, and Butler played out the 2016 season on a base salary of $600,000 in the final year of his deal.
In addition to Butler, the Patriots are not expected to trade backup quarterback Jimmy Garoppolo this offseason, and he is expected to go to training camp with them, league sources told Schefter.
ESPN's Mike Reiss and Mike Triplett contributed to this report.