The 32 year-old midfielder is out of contract after leaving West Ham at the end of last season having struggled to make an impact in his second spell at Upton Park, making just 13 starts.
Cole began his career at West Ham before spells at Chelsea and Liverpool followed, with the England international spending most of the 2011/12 season on loan at Lille.
If the move to Villa Park goes through, Cole will become the second player to sign for Villa during the close season, after defender Philippe Senderos joined on Thursday after being released by Fulham.
Cole announced at the end of the season that he would be leaving the Hammers and was keeping an open mind regarding his future options.
He said at the time: "My relationship with West Ham is great, but obviously it didn't work towards the end of this season and I didn't play enough.
"So, I'm just going to take my time over the summer and find the right fit.
"At this stage of my career, finances are not important for me. I want to enjoy my football, I want to compete and I just want to play more."