The 32-year-old midfielder is a free agent after leaving West Ham at the end of the season, where he made only 13 starts in his second spell with the club.
Cole reportedly turned down bigger financial offers from overseas clubs to try and stay in the Premier League with Paul Lambert's side, with the Scotsman keen to add some experience in the centre of the park.
He began his medical on Monday evening and, with that hurdle now overcome, could be unveiled as a Villa player this week.
Villa have made one signing already this summer, bringing in Switzerland centre back Philippe Senderos after his release from relegated Fulham, having spent the second half of the season on loan with Valencia.
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."