Contact tracing is being done to determine whether any other players or staff members will be required to enter the NBA's health and safety protocols, which means a minimum of a seven-day quarantine with no positive test.
The Athletic first reported Kleber was entering the league's health and safety protocols.
Three key Mavericks -- backup point guard Jalen Brunson, starting small forward Dorian Finney-Smith and starting shooting guard Josh Richardson -- entered the health and safety protocols after one of them tested positive on Friday. Those players remained quarantined in Denver, where the Mavs defeated the Nuggets in overtime on Thursday night.
It has not been specified which of that trio tested positive.
Kleber started and played 24 minutes in the Mavs' home win over the Orlando Magic on Saturday night.
Dallas' next game is scheduled for Monday night at home against the New Orleans Pelicans.
Pelicans head coach Stan Van Gundy told reporters that he "absolutely" expects the game to be played and cited a similar scenario that unfolded Saturday, when the Philadelphia 76ers had only seven healthy players for their game against Denver.
"If they played that Philly game, I think they're going to play every game they can play," Van Gundy said Sunday.
When asked whether he thinks games should be played with short-handed rosters, Van Gundy deferred to the NBA and the NBA Players Association.
"It's not for me to say. The NBA and the Players Association got together and decided how they were going to handle this year," he said. "A lot of it is about economics and a lot of it is about putting our product out there on TV in front of fans, which I think is great. ... I don't have as much knowledge of the situation as Adam Silver and his staff do or all the teams do, so I won't say what people should or shouldn't do. I'm just going to follow what they tell me to do."
ESPN's Andrew Lopez contributed to this report.