Indians manager Terry Francona announced the move before Saturday night's game against Tampa Bay. He met with Axford prior to batting practice.
Axford is 1-3 with nine saves and a 4.91 ERA in 17 games in his first season with the Indians. He has really struggled in the last week, and was pulled in the ninth inning of Cleveland's 6-3 win on Friday night.
''We told him, for now, we're going to kind of get you out of that role and try to get you in some situations where we can get him on a roll again,'' Francona said.
''I told him numerous times, because I mean it, the goal is to get him back as our closer, and I think he understands that. I don't want him to do it in like two days because I don't want some Band-Aid. I want him when he gets it, to take it and stay there,'' he said.
''The one thing I assured them is that when we get to the ninth, whoever's pitching, we're not going to have four guys warming up because we think you're good,'' Francona said.
Axford was signed to a one-year, $4.5 million contract as a free agent in December after pitching for Milwaukee and St. Louis last season. The one-time Brewers closer had 105 saves from 2010-12, and converted 49 straight save chances during a stretch.
The Brewers traded an ineffective Axford to St. Louis in late August. Axford became a reliable reliever for the Cardinals down the stretch and pitched for them in the World Series.
The Indians hoped the 31-year-old righty would do well replacing former All-Star closer Chris Perez, now with the Los Angeles Dodgers.
But Axford's rough patch started last Sunday, when he gave up a three-run homer to Dayan Viciedo in the ninth inning of a 4-3 loss to the Chicago White Sox.
The next day, Eduardo Escobar homered in the 10th off Axford and lifted Minnesota to a 1-0 win.
Axford's latest tough outing came Friday night. He entered in the ninth with a 6-2 lead, but gave up David DeJesus' RBI double with two outs and then walked Ben Zobrist to load the bases. Allen replaced Axford and got a fly ball from James Loney on a 3-2 pitch for his first save.