The three-time Cy Young winner continues to rehab from a lower back strain that has sidelined him since July 23. The Dodgers open a four-game series on Monday against the Pirates.
Dodgers manager Dave Roberts told reporters Sunday that the 29-year-old Kershaw is expected to pitch a rehab game next weekend with either Double-A Tulsa or Triple-A Oklahoma City.
"Once he passes that (simulated game) marker right there, then we'll see where he goes," Roberts said. "But it's probably going to be a (rehab) start either in Tulsa or OKC."
Kershaw threw off flat ground on Sunday with positive results following a 38-pitch simulated game Wednesday at Dodger Stadium.
Kershaw, who was on his way to a runaway Cy Young season when he encountered tightness in his back after the second inning against the Atlanta Braves on July 23, was 15-2 with a major-league-best 2.04 ERA with and 168 strikeouts in 141 1/3 innings in 21 starts before the injury.
Jankowski, 28, was optioned to Triple-A Oklahoma City.
Jankowski has gone 24-16 with a 3.46 ERA in 233 games (28 starts) in six minor league seasons in the Houston organization, striking out 509 in 421 innings and limiting opponents to a .228 batting average.
Jankowski made his major league debut on May 24 against the Detroit Tigers, and allowed six runs in 4 1/3 innings in three games with the Astros this season before being designated for assignment on Monday. He spent the majority of the 2017 season with Houston's Triple-A Fresno, going 2-3 with a 5.13 ERA in 37 games.
On Friday night, the Dodgers acquired Granderson and cash considerations from the Mets in exchange for a player to be named or cash considerations.
Rhame, 24, went 0-2 with two saves and 4.31 ERA in 41 games this year with Triple-A Oklahoma City and has posted a 10-18 record with 29 saves and a 3.11 ERA in 210 minor league games in the Dodgers' organization.