According to The Athletic, the 30-year-old is being looked at as a replacement for the injured Wesley.
The Brazilian has been sidelined for the remainder of the season with an injury and Smith needs to bring in reinforcements.
Rodriguez has shown his quality in the Premier League over the years and he could prove to be a solid short-term addition for Villa.
The Burnley striker has good link-up play and he can score goals.
Aston Villa cannot play the remainder of the season without a recognised striker and the interest in Rodriguez makes a lot of sense.