The contract will be the second-largest ever given to a free agent reliever, only behind Aroldis Chapman's five-year, $86 million deal with the New Yankees last week.
The 29-year-old Jansen recorded 47 saves with a 1.83 ERA in 71 appearances for the Dodgers in 2016.
Jansen became the all-time saves leader in Dodgers history on June 20, passing Eric Gagne's previous record of 161. Jansen finished the season with 189 career saves in 409 games over his seven seasons with the Dodgers.
The Dodgers also are close to retaining third baseman Justin Turner, according to the Los Angeles Times.
Turner's deal would be for a reported four years and $64 million.
Turner, 32, hit .275 last season and set career highs in runs (79), hits (153), doubles (34), triples (3), home runs (27), RBIs (90) and games played (151) while coming off knee surgery.