Hample, who attended Sunday's Dodgers- Reds game in Cincinnati, claimed in a series of tweets that he asked Kershaw for one of the Father's Day baseballs, only to be turned down by the ace pitcher.
Hample, attempting to make a logical argument, also jabbed at Kershaw, saying that the star left-hander's "next paycheck shouldn't mean all that much to him, right?"
Hample made headlines in 2015 when he caught Alex Rodriguez's 3,000th hit -- a home run -- at Yankee Stadium and negotiated a compensation package from the Yankees in exchange for returning the ball to the now-retired slugger.
As part of the agreement, the Yankees donated $150,000 to a charity that Hample supports.
Hample, 39, also drew the ire of the 82nd Airborne Division last year when he sneaked into a game at Fort Bragg that was meant exclusively for military personnel.