The Twins cut the 29-year-old Rincon after he put up a 5.44 ERA over the last two seasons. Before 2007, though, he was a reliable late-inning arm, posting a 2.93 ERA between 2003 and '06.
Manager Eric Wedge sounded encouraged by the reports on Rincon, if not the numbers.
"He's showing a good arm, a good slider," Wedge said. "He has tremendous late-inning experience, and this is a good opportunity to get him in the mix."
But the Indians have another roster move looming, as they'll need a starter for Saturday's game against the Rays.
Fausto Carmona (left hip strain) threw another bullpen session on Tuesday and should be ready for a simulated game Thursday, followed by a rehab assignment. But he won't be ready to rejoin the Indians until after the All-Star break, so a spot starter is needed for at least one outing.
Contrary to previous reports, that starter won't be veteran Jeff Weaver, who was signed to a Minor League deal last week. The Indians want Weaver, who was cut by the Brewers from Triple-A Nashville on June 11, to get some innings under his belt before he's considered as an option.
Left-hander Dave Huff (2-2, 3.79 ERA), who jumped from Double-A Akron to Buffalo earlier this season, is also not an option, since the Indians don't want to over-accelerate his development.
That leaves right-handers Matt Ginter (6-6, 4.27) and Dan Reichert (1-3, 4.40) and left-hander John Halama (4-2, 5.66) as possibilities for the Saturday start.
Or, as Wedge described the candidates, "guys who have been around for a while."
Anthony Castrovince is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.