Sweeney, a 38-year-old left-handed hitter, led the Majors with 24 pinch-hits last season. He had 23 hits, including eight doubles and two homers, in 90 at-bats and drove in 10 runs for the San Francisco Giants, and went 9-for-33 with three RBIs after being traded to the Dodgers on Aug. 9.
"We'll hopefully get everything resolved tomorrow," Sweeney's agent, Barry Axelrod, said Sunday.
Axelrod said a key component was getting a Major League contract, and not a Minor League deal.
Sweeney, who can play first base and the outfield, has a .260 career batting average while playing 13 seasons with St. Louis, San Diego, Cincinnati, Milwaukee, Colorado, San Francisco and Los Angeles.
Sweeney's talks with the Dodgers were first reported by ESPN.com.
Los Angeles opens Spring Training at Vero Beach, Fla., for the final time this week. In the Dodgers' first season under manager Joe Torre, they've added center fielder Andruw Jones, pitchers Hiroki Kuroda, Chan Ho Park and Tanyon Sturtze, and backup catcher Gary Bennett.