Wedge has talked with League and plans to stick with the 28-year-old, who is filling in for injured closer David Aardsma this season.
"It's not so much re-evaluation as just finding out where he is. You've got to be careful how you phrase that," Wedge said. "I don't want to mislead anybody. I talked to him a little last night, and today I'll give him the day off because he needs it, physically, and it'll be good for him mentally, too.
"We'll see how he feels tomorrow and go from there. We'll take it day by day. He's still our closer. If we need somebody to close today, we'll go with Jamey Wright. He's been in that setup role, and you always need somebody to pick you up when your closer isn't available. Today is a good day for [League] to watch a ballgame and go from there."
Wedge feels that League has been a little up with his sinker the past few games, but that he has also been the victim of some difficult circumstances.
"The ball is coming out of his hand good. The action is good," Wedge said. "I think everybody has to understand, it hasn't all been just him. There's a couple of plays we haven't made behind him, a couple of ground balls that found holes. You just have to deal with that.
"Granted, he hit a couple of guys the other day, and that made things difficult for him. But if you look at us as a ballclub right now, we were in position to win that last extra-inning game at home, and that would have been four straight series. We were in great position to win two of three in Baltimore and great position to win last night.
"We're sitting here on a six-game losing streak, but we could easily be sitting here with five series wins in a row and 1-0 in this series. That's how fragile it is and how close we are."
League wasn't dwelling on his recent struggles when he was on the mound on Friday as the Indians scored three times to overcome a two-run deficit, a rally capped by Travis Hafner's two-run walk-off homer.
"The last thing I'm thinking about is [previous games]," League said. "But after the game, it's a different situation. It's tough and hard, but you've got to do it. We play every day, and it's a long season. You have to stay positive."