The prospect is right-hander Zach Eflin, according to multiple media reports.
The Dodgers will also send about $32 million to San Diego to cover part of Kemp's remaining $107 million salary, a source told ESPN.
The trade helps alleviate a logjam of outfielders in the organization. Andre Ethier and prospect Joc Pederson will vie for playing time, with Yasiel Puig holding the starting job in right field, and Carl Crawford in left. The Dodgers could also use role player Scott Van Slyke in the rotation.
The Dodgers' new front office has been looking to move bulky long-term contracts and to improve the atmosphere in the clubhouse, which was described as "dysfunctional" by several team sources last season. The glut of outfielders became a source of tension at various times throughout the season.
Kemp, 30, looked rejuvenated this season after the All-Star break, hitting 17 of his 25 home runs and leading the major leagues in slugging percentage over that time period. Kemp finished second in voting for the National League MVP award in 2011, when he batted .324 with 39 home runs and 126 RBIs. He signed an eight-year, $160 million contract that winter.
Injuries cost him more than 150 games over the last three seasons. He underwent a major shoulder operation after playing 106 games in 2012 and ankle surgery after playing 73 games in 2013.
Kemp tweeted his thanks to Dodgers fans on Thursday:
It is with a lot of emotion that I say goodbye to the city and the fans who have been there since the... http://t.co/IbyhZWQ55B
- Matt Kemp (@TheRealMattKemp) December 11, 2014
The agreement is the latest in a flurry of moves for the Dodgers, who, in a busy Wednesday, acquired second baseman Howie Kendrick, infielder Enrique Hernandez, reliever Chris Hatcher and minor leaguer Austin Barnes. Los Angeles also added veteran shortstop Jimmy Rollins and agreed to a four-year deal with pitcher Brandon McCarthy, sources told ESPN.
"It's been very exciting and sleep-deprived," Dodgers senior vice president of baseball operations Josh Byrnes said.
"I can't talk about much of the details, but the objectives were to preserve the top three prospects, win now, get a little more prospect depth, change some components of the roster, but maintain a lot of the strengths of the roster. So, it's been a balancing act. We've still got work to do, but it's been fun."
Since taking over on Oct. 14, president of baseball operations Andrew Friedman and his general manager Farhan Zaidi have made 10 different trades (three Wednesday alone), claimed two players off waivers and have signed one free agent (McCarthy), according to ESPN Stats & Information.
Information from ESPN The Magazine's Buster Olney, ESPN.com's Jerry Crasnick and ESPNLosAngeles.com's Mark Saxon was used in this report.