Gronkowski was injured on the Patriots' eighth offensive play of Sunday's 41-28 victory over Houston. He previously missed five regular-season games and is done for these playoffs.
The person spoke on condition of anonymity because the team has not released details of the injury.
"I'm not sure," coach Bill Belichick said after the game when asked if Gronkowski had broken his arm.
Asked if Gronkowski had been taken to a hospital, Belichick said, "Look, I just walked off the field."
He also said "he wouldn't have played if he wasn't ready" and that "the doctors handle the medical decisions."
Rookie defensive end Chandler Jones hurt his ankle later in the game.
Gronkowski had broken his left forearm while blocking for an extra point near the end of New England's 59-24 victory over the Indianapolis Colts on Nov. 18 and underwent surgery. He missed five games and returned for the regular-season finale, a 28-0 win over the Miami Dolphins.
He was reinjured just 6:44 into Sunday's game on a deep pass down the right sideline from Tom Brady. Gronkowski fell out of bounds as the pass dropped incomplete.
He sat on the bench, writhing in pain, while talking to team doctor Thomas Gill before heading to the locker room.
Brady provided no information on the condition of one of his most important teammates.
"I don't know anything about that," Brady said. "I haven't heard anything."
Gronkowski's teammates said that the team would miss him but that other players have to contribute.
"It's hard to replace a player like him because he's a freak of nature," tight end Aaron Hernandez said, "but everyone has to step up and everyone has to keep making plays so we can keep it rolling. (He) definitely helps me out because so much attention is on him.
"It's a big loss and you can't replace a player like him."
Last season, Gronkowski's status was listed as questionable for the Super Bowl just 48 hours before the New York Giants 21-17 victory. He had suffered a high sprain to his left ankle two weeks earlier in the Patriots' 23-20 win over the Ravens in the AFC championship game. That hampered him in the Super Bowl in which he had just two catches for 26 yards after a season in which he had 105 receptions, 15 of them in the other two postseason games.
After the season, he had arthroscopic surgery on the ankle.
On Sunday, backup tight end Michael Hoomanawanui got more action once Gronkowski left.
"I just saw him a little bit ago," Hoomanawanui said. "A guy with that much talent, it stinks. There's no other way to put it. I went down with a knee injury last year that ended my season and you never want to see that for anyone, for an opponent, but let alone a guy that you spend a lot of time with each and every day."
Asked if Gronkowski would miss the rest of the postseason, Hoomanawanui said, "I haven't heard yet. He looks like he was hurting. I'm sure we'll find out here soon enough."