The Indians have 10 days to either trade or release Lowe.
One of Lowe's former teams, the Boston Red Sox, are debating whether to add the 39-year-old, a team source told ESPNBoston.com's Joe McDonald. Lowe spent eight seasons with the Red Sox and was a key member of their World Series championship team in 2004.
Lowe was a guest on Boston sports radio station 98.5 The Sports Hub on Thursday and acknowledged that the Red Sox would be his preferred destination because of the respect he has for the franchise.
Red Sox manager Bobby Valentine was asked Thursday if there have been internal discussions about the possibility of Lowe returning to Boston.
"Well, I bet we're talking about him," Valentine said. "If he's available and someone might think he can help our situation, I bet he's being talked about and evaluated."
Lowe gave up seven runs on eight hits, two walks and a balk over 2 1/3 innings in Cleveland's 8-3 loss Tuesday night. He had lost four straight, had a 15.88 ERA in his past three starts and an 8.80 ERA in his past 12.
Lowe began the season 6-1 with a 2.05 ERA, but is 2-9 since with his ERA ballooning to 5.52. The 16-year veteran is 174-156 with 85 saves in his career with Cleveland, Atlanta, the Los Angeles Dodgers, Boston and Kansas City.
Rookie right-hander Corey Kluber will take Lowe's roster spot and make his first big league start Thursday night against the Royals.
Information from ESPNBoston.com's Joe McDonald and The Associated Press was used in this report.