The deal, first reported by the Atlanta Journal-Constitution, will become official once both players have completed their physicals.
Olivera, a 31-year-old Cuban defector, played just six games this season before being hit with an 82-game suspension without pay for violating the league's domestic-violence policy. The suspension is set to end Monday.
The Braves acquired Olivera last summer in a trade with the Los Angeles Dodgers, who signed him to a six-year, $62.5 million contract before the 2015 season. He is hitting .211 with no homers and 2 RBIs in six games this season.
Kemp, 31, hasn't lived up to his All-Star form since being traded to San Diego two seasons ago. This season, he is hitting .265 with 23 home runs and 69 RBIs, but has posted a .285 on-base percentage.
He signed an eight-year, $160 million deal with the Dodgers in 2012.
Saturday's trade is the latest among a series of deals the Padres have made in the past two months as they move to rebuild.
The team also sent starting pitcher James Shields to the Chicago White Sox for starter Erik Johnson and minor league shortstop Fernando Tatis Jr., and sent closer Fernando Rodney to the Florida Marlins for right-handed prospect Chris Paddack.