The guard underwent X-rays after practice that revealed the injury. The Lakers said they are evaluating treatment options for the guard.
Hart, who is a right-handed shooter, suffered the injury to the ring finger on his left hand.
He will return to Los Angeles on Thursday to have his hand examined, with an update on his treatment expected to follow.
Hart is averaging 6.6 points on the season. He started the past 10 games. He has four double-doubles as a starter.
"[Hart is] very important," Lakers rookie guard Lonzo Ball said. "Double-double pretty much every time he steps out there, one of our best defenders and also rebounds a lot. He helps a lot, especially defensively."
Kuzma, who likes to give his friends on the team a hard time on social media, wrote, "You posted this like your career ending boy if you don't relax and go play fortnight lol."
Nance, who is now with the Cleveland Cavaliers, wrote, "I was back in 3 weeks man ... Dramatic!!"