Cole is a free agent after his contract with Chelsea expired and had been heavily linked with a move to the Stadio Olimpico in recent days.
The 33-year-old now seems to be on the verge of completing that switch, after Roma posted a picture on their official Twitter account announcing Cole's arrival.
An image of Cole was accompanied by the message: "Welcome to Rome, Ashley!"
Cole announced his retirement from international football in May after missing out on a place in Roy Hodgson's England squad for the FIFA World Cup in Brazil.
A veteran of 107 caps, he claimed eight major honours during his time at Chelsea, including the 2011-12 UEFA Champions League.
In addition to winning three Premier League titles, two with Arsenal and one with Chelsea, Cole has also won the FA Cup on seven occasions.
However, he fell out of favour under Jose Mourinho last season, making just 15 Premier League starts as Cesar Azpilicueta emerged as Chelsea's first-choice left-back.
