Manager Eric Wedge made the decision Friday, a day after Betancourt had to be replaced in the ninth inning of a 4-2 win over Oakland.
"I talked to Betancourt today," Wedge said before a 4-3 loss to the Reds. "I explained to him what I'm seeing and where he's at. He understands. I think we're capable of using any of those guys."
Borowski has been out since April 15 with a strained right triceps. He threw 30 to 35 pitches without problem Friday in a simulated game. The Indians plan to have him pitch in a minor league game next week as part of a rehabilitation assignment.
Wedge would like to see Borowski pitch a couple of times in the minors before he's activated.
Betancourt tied for the AL lead in holding leads last season, when he was a key part of Cleveland's bullpen. He moved into the closer's role when Borowski got hurt, and converted four of five save chances even though he struggled. On Monday, he gave up three runs in the 10th inning of a scoreless game.
Betancourt was 1-2 with a 7.31 earned run average after he failed to close out Thursday's game. He loaded the bases with one out in the ninth on a pair of hits and a walk.
Masa Kobayashi, signed as a free agent from Japan last November, threw a run-scoring wild pitch before closing it out for his first major league save. Kobayashi saved 227 games in Japan, one of only three pitchers there to reach the 200-save mark.
"I look at it like we have four guys I feel comfortable using in the seventh, eighth and ninth innings, depending on the matchup and availability," Wedge said. "That's how I'm goring to play it out. I'm not going to say somebody is or somebody isn't [a closer]. I feel we can be comfortable using all four of those guys."
Wedge wasn't comfortable picking one player as the closer.
"I think the worst thing you can do is anoint somebody that one day later," he said. "We need to have more to work off. We're still optimistic Joe is going to be back sooner than later. It's not an indefinite situation."
Heading into the intrastate series, the Indians had won eight of 10, vaulting them into first place in the AL Central.