Paul Lambert’s side are in desperate need of more firepower, having scored only 17 goals in 21 Premier League games this season.
The Villans are just a point above the relegation zone and the Scot knows he must improve his side's attacking potency if they are to avoid a very difficult second half to the campaign.
Villa were linked with Robbie Keane, but the Republic of Ireland international has now agreed a new contract at LA Galaxy, and have instead turned their attention to Graham.
The 27-year-old, who has also been linked with Reading and Sunderland, has often found himself on the bench at the Liberty Stadium this season, following the summer arrival of Michu and, although the Welsh club are keen to hang on to him, a big money bid could tempt them.
And talkSPORT understands Lambert could be ready to spend almost all his available funds and make a £12million swoop for Graham.