The Villans are in desperate need of some firepower having scored just 17 goals in their 21 league matches so far this season, which as a result has left them a point above the relegation zone in 16th place in the table.
Villa were linked with LA Galaxy striker Robbie Keane, who previously had a loan stint at Villa Park, but the Republic of Ireland international recently penned a new deal with the MLS outfit.
Now it appears that manager Paul Lambert has turned his attention to Graham, and could use nearly all of his available funds by bidding up to £12m for the forward, according to talkSPORT.
Both Reading and Sunderland are rumoured to be interested in the 27-year-old, who scored a late equaliser in the Swans' 2-2 FA Cup draw with Arsenal on Sunday, but Michael Laudrup has since stated that he could keep hold of Graham.
Graham has scored six goals in all competitions so far this season, having mainly played second fiddle to star striker Michu.