Those outings have prompted manager Don Mattingly to consider swapping League and setup man Kenley Jansen.
"Obviously I can't say I haven't given thought to it," Mattingly said before Wednesday's game against the D-backs. "The games that Brandon has given up runs have been games he wasn't necessarily closing. But Kenley is throwing the ball good. I've given thought to it."
Are the Dodgers close to making a change?
"I'm just still thinking about it," Mattingly said. "We still talk about it and go back and forth. It's like anything else -- all of a sudden we flip it and then we got a mess. Not that we're not in a mess already a little bit. But Brandon is actually throwing the ball OK. He's not walking a bunch of people. In save situations, he's been really good."
League has converted eight of nine save opportunities. His past two outings were not save situations, but League said he has the same mindset each time he takes the mound.
"You have to do your job and hold the score," said League, who signed a three-year, $22.5 million contract in the winter.
League's lone blown save came against the Mets on April 24. He recorded a save the next night, though.
"I don't feel like Brandon has done anything really wrong," Mattingly said. "He gives up the home run to Quiroz and the home run [Tuesday] night. And even the blown save, we feel like if we make plays for him he gets that one. For me right now, I'm going to go with Brandon."
Austin Laymance is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.