The injury appeared to have occurred during a one-on-one blocking drill in the Patriots' full-pads practice. Henry later left for the locker room with a member of the team's medical staff and eventually had an MRI.
The injury is "not serious," the source told Schefter, but there is no definitive timetable for Henry's return.
"Hunter has participated in everything he's been able to since he's been here [in the spring]. He's gained a lot of experience," coach Bill Belichick said Tuesday morning. "Hunter's a smart guy. He'll come back when he can come back."
The Patriots host the Washington Football Team on Thursday in the preseason opener for both teams.
Henry, 26, signed a three-year, $37.5 million contract with the Patriots after five years with the Los Angeles Chargers. New England's multiple tight end set includes Henry and fellow free-agent signee Jonnu Smith (four years, $50 million), formerly of the Tennessee Titans.
The 6-foot-5, 250-pound Henry has 196 career receptions for 2,322 yards and 21 touchdowns.
Henry entered the NFL as a second-round draft pick of the Chargers in 2016, and played in 55 of a possible 80 regular-season games with the team over five seasons. He missed the entire 2018 regular season after tearing an ACL in offseason practices, returning only for one playoff game.
ESPN's Mike Reiss contributed to this report.