The report claims that the Villans are looking at the Icelandic midfielder as a possible alternative to fellow target Mile Jedinak, though they face competition from Championship duo Derby County and Norwich City.
Villa, under the guidance of owner Tony Xia and newly-appointed boss Roberto Di Matteo, have made a poor start to the season with defeats against Sheffield Wednesday and Luton Town in the EFL Cup.
It was always going to be challenge for Villa following their disastrous Premier League campaign last season, what with the club having failed to record a competitive win since last February.
But the club have been active in the transfer market this summer, and they are now looking to add some steel following the recent addition of defender James Chester from West Bromwich Albion.
Crystal Palace’s Jedinak is seemingly the club’s number one target, yet a failure to lure him to the Championship will see them turn their attentions to Cardiff’s Gunnarsson.
The Iceland international played a key role during his nation’s run to the Euro 2016 quarter-finals, prompting heavy interest from Derby and Norwich as the duo look to bolster their ranks.
Villa have the means to beat both of their fellow expected promotion rivals to Gunnarsson’s signature, though they may have to move quickly as teams become even more desperate ahead of the transfer deadline.
In the same report, Chelsea's Patrick Bamford and Leicester City's Ritchie De Laet are also named as targets for Villa.