The 30-year-old Frenchman is out of contract in the summer and could reportedly be available for a cut-price fee this month.
Congre, 30, originally joined Montpellier from boyhood club Toulouse in June 2012 and has since made 132 total appearances for La Paillade, while recording four goals in the process.
The former France Under-21 international has started all but one of the team’s 18 Ligue 1 games this season but is out of contract in the summer and still to begin talks over a new deal.
Villa boss Remi Garde reportedly admired the veteran centre-back during his time as Lyon manager between 2011 and 2014 and is now said to be eager to bring him to England this month in an attempt to stabilise his side’s struggling defence.
Although primarily a centre-half, the 6ft 1in stopper can play anywhere across the backline and would not set Villa back too much in terms of a transfer fee given his contract status.
Whether Garde can convince him to come to the club considering their current predicament – 11 points from safety at the bottom of the Premier League table – obviously remains to be seen, but the Frenchman does not believe that will be a problem when it comes to attracting potential signings during the winter transfer window.
“Of course, the situation is not the best one you could dream to speak with players about,” Garde told Villa’s official website this week. "But what I have done already doesn't show me that nobody wants to come to this football club.
"You have a lot of players who are clever or smart. They might look at the next two, three or four years, the best project may be this one, despite what can happen.
"That's because of many reasons and many things around the football club. Of course, it's about the short-term but then you can look and say 'I understand what they want to do, I want to be part of it.'"
Villa could also have a defensive spot in their squad opening up in the near future, with Liverpool loanee Tiago Ilori reportedly set to have his season-long spell terminated early after failing to see any first-team action for the club over the first half of the campaign [h/t Telegraph].