The Reds would play Choo in center field, a position he has played only 10 times in his Major League career.
Cincinnati is willing to move Gregorius because he is blocked at the MLB level by young shortstop Zack Cozart, and there is no sign of an opening in the near future. They have been pursuing outfielders on the trade and free agent market and recently resigned Ryan Ludwick to a two-year contract.
The Indians have been shopping Choo and shortstop Asdrubal Cabrera, and if they acquire Gregorius he could be moved to Arizona in a potential Justin Upton trade.
Cleveland has been both shopping current veterans as well as shopping for free agents, and they recently signed Mark Reynolds to a one-year contract.
The Reds are a potential suitor for free-agent center fielder Michael Bourn, and if they acquire Choo it would further shrink the market for Bourn.
Cincinnati has been linked to Dexter Fowler in trade rumors as well, but it is reported that Fowler is unlikely for them at this point.
The Reds finished in first place in the National League Central in 2012, losing in 5 games in the National League Division Series to the eventual World-Champion San Francisco Giants.