The Villains are desperate to add a striker of quality in the January transfer window.
They lost their attacker Wesley due to an ACL injury, which keeps him out for the rest of the season.
The Athletic suggests that Villa are seeking loan signings this month.
It could mean a move for Rodriguez, but Burnley are likely to want a permanent deal.
Both teams are battling relegation and doing Villa favours will not be on Sean Dyche's mind.
The Clarets would part with Rodriguez, but only if they get a good fee they can use to invest in their team.