Klay won the NBA championship last year with the Warriors and his team is on another historic pace this season.
For the White Sox in 2015, Trayce hit .295 in 135 plate appearances with five home runs. In December, he was dealt to Los Angeles as part of a three-team deal.
Thompson said his competitiveness motivates him to work during Spring Training -- and that comes from his family. His father, Mychal, played in parts of 12 seasons in the NBA and won two NBA titles with the Lakers' Showtime teams in 1987 and '88.
"It doesn't matter who's in front of you, if you want to be the best you've got to be competitive and you have to be able to prove yourself out there," Thompson said. "I'm definitely a very competitive person."
Thompson is still competitive with his older brothers.
"If I lose to them in something, it ruins my day," he said. "Klay, he can't beat me in golf. The competitive edge isn't really there. He's beat me maybe once in 30 times."
This spring, Trayce has struggled at the plate, batting .208 with two home runs and eight RBIs as he tries to prove last year wasn't a fluke.
"These coaches haven't really seen me firsthand before, so I'm sure they're trying to see what type of player I am," he said. "You want to make a strong impression, but at the same time, you have a season to prepare for."
Instead of focusing on his struggles at the plate, Thompson is focusing on learning as much as he can from players like Chase Utley.
"To watch kind of an old-school player like Chase, he goes about the game in a very hard-nosed way, doesn't take any plays off," he said. "That's something that I'll definitely take away."
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Thompson could have more time to learn from Utley and others in the clubhouse as Spring Training begins to wrap up. News broke earlier this week that outfielder Andre Ethier suffered a broken leg that will keep him out 10-14 weeks.
The news greatly affects Thompson, who goes from a likely casualty of the final round of cuts to being in the running for playing time in Ethier's absence.
Dodgers president of baseball operations Andrew Friedman is confident in the depth of his roster, including Thompson, in the wake of Ethier's injury.
"I don't know exactly what we'll do, but I know we have a lot of talented guys that we'll talk through and debate," Friedman said.
Prior to a round of cuts Monday, manager Dave Roberts took Thompson aside to let him know his hard work has not gone unnoticed, despite his lack of production at the plate.
Roberts is confident Trayce can produce should the club decide to give him playing time along with Scott Van Slyke and Carl Crawford.
"After you put the ball in play, he's not punching out a whole bunch of times. You can only control so much, so I think he's held his own," Roberts said.