6:35pm: Jayson Stark at ESPN is reporting that the move is official. Padilla will sign a minor-league deal and make at least one start in the minors before he's called up to the Dodgers' rotation. He'll be owed the prorated portion of the league minimum–around $100k.
It's a cheap way for the Dodgers to add depth, since they'll only be responsible for the pro-rated MLB minimum, with the Rangers paying Padilla the rest of his $12MM salary.
4:36pm: Dylan Hernandez of the LA Times hears that the Dodgers are "very" close to adding Padilla.
12:45pm: Jon Paul Morosi of FOX Sports hears that the Dodgers are "moving closer" to an agreement with Padilla.
12:19pm: ESPN.com's Jayson Stark hears that Vicente Padilla has become "a consideration" for the Dodgers now that a liner to Hiroki Kuroda's head has further depleted the team's rotation. The Dodgers are talking with Padilla's agent, Paul Kinzer, and we could hear of a deal this afternoon. The Dodgers, who would only be responsible for about $100k if they add Padilla, had uncertain interest as recently as yesterday but they're now considering him.