Major League Baseball declared Rodriguez, 21, a free agent on Monday. The Reds have not confirmed the deal.
"He is a player of interest, but we do not have a deal," Reds general manager Walt Jocketty told the Cincinnati Enquirer on Monday night.
Rodriguez batted .265 last season with Isla de la Juventud in Cuba's Serie Nacional, where he won a Gold Glove in 2014 and 2015.
The Reds already have multiple shortstops in their organization, including Zack Cozart, who is expected to be ready for 2016 after a season-ending knee injury last year, Eugenio Suarez and prospect Jose Peraza, who was acquired in the blockbuster trade involving Todd Frazier.