The veteran left back fell behind Cesar Azpilicueta in the pecking order at Stamford Bridge last season and is now keen to push for glory in Serie A with his new club after making the switch at the end of his contract.
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Cole, who has signed a two-year deal with the Giallorossi, told reporters: "I've always had a good relationship with Mourinho but the years go by and I was no longer useful for his project at Chelsea. At that time my agent told me about Rome and how badly they wanted me for their project [at Roma].
"As soon as I spoke to [coach] Rudi Garcia and I saw how much they wanted me here at Roma, I jumped at the chance. I came here to play in the Champions League. Roma is an ambitious club, it's a challenge for me and I'm prepared to fight for my place. Hopefully we can do something special here next season and win Serie A."
David Beckham is the only other Englishman to have plied his trade in Italy in recent years - the midfielder having spent two loan spells at AC Milan - and Cole feels that there is a reluctance for his countrymen to move out of their comfort zone at home.
"English players are afraid of going abroad and often make the easy choice of staying at home," he observed. "This is a chance for me to get out of London, try a different language, culture and way of living. I'm not afraid of pressure.
"I feel fit, I kept myself busy in preseason but I still have a bit to go. Hopefully I'll be fully fit when I come back from [Roma's preseason tour in] America."