Burnett's contract will include an opt-out clause on May 1 if he isn't added to the roster, per Rosenthal.
The Dodgers are currently carrying just one left-handed reliever in J.P. Howell, with both Adam Liberatore and Luis Avilan optioned to Triple-A Oklahoma City.
Burnett hasn't pitched since the 2014 season and in the last three years combined has pitched a total of 16⅓ innings in the majors and minors. His most recent recovery was Tommy John surgery in 2014, 10 years after his first Tommy John procedure.
Burnett also missed time in 2013 with forearm irritation and a left elbow impingement.
The left-hander was one of the most reliable relievers in the game before getting hurt, putting up a 2.85 ERA while averaging 71 games, 58 innings, 49 strikeouts and 20 walks per year from 2009-2012. Burnett, 33, has held left-handed batters to hitting just .228/.293/.336 in his career.
Burnett was in Nationals camp as a non-roster invitee this spring and had nine scoreless appearances, pitching 8⅔ innings with seven strikeouts and only one walk. But after he was one of the final roster cuts and didn't make the team, Burnett opted out into free agency.