The 33-year-old is a free agent following his release by the Blues earlier this summer and has been linked with a number of teams, both in the Premier League and in Europe.
However, he looks set to sign for Roma after flying out to the Italian capital, with the Serie A side posting a photo on their official Twitter feed, with the message: "Welcome to Rome, Ashley!".
According to the Italian media, Cole is poised to put pen to paper a two-year deal worth around £1.82m, with an option for a third season as Roma look to strengthen their squad ahead of their return to the UEFA Champions League.
Cole signed for Chelsea in a controversial move from rivals Arsenal in 2006 for £5m and went on to make over 300 appearances for the club.
He lifted the Premier League under Carlo Ancelotti in 2010, before clinching the 2012 Champions League under Roberto Di Matteo, as well as the 2013 Europa League with Rafa Benitez, and five FA Cups.
Cole has also made 107 appearances for England, but announced his retirement from international football ahead of the World Cup after he was left out of Roy Hodgson's squad for the finals in Brazil.
Roma are also close to signing Fenerbahce's Salih Ucan, with the 20-year-old midfielder set to sign a two-year loan deal with an option to buy for a reported fee of around £8.73m.
See Roma in Champions League action next season on Sky Sports 5 - a new channel dedicated to European football.