Former England midfielder Cole is a free agent after his contract at West Ham expired.
Villa had been strongly linked with the 32-year-old and a deal now seems close to completion.
The Midlands outfit posted a picture of Cole in Villa training gear on their official Twitter account on Tuesday, accompanied by the message: "Joe Cole at Bodymoor Heath (Villa's training ground) having completed a medical this morning."
After lengthy spells with West Ham and Chelsea earlier in his career, Cole has switched clubs on a regular basis in recent seasons.
He joined Liverpool in 2010, but was soon loaned out to French side Lille, before rejoining West Ham in January 2013.
Cole made only six Premier League starts last season, featuring as a substitute on 14 further occasions.
