The ex-England international signed a two-year deal with the Serie A runners-up last week, and has set his sights on glory both domestically and in the UEFA Champions League.
With Mourinho favouring Cesar Azpilicueta as Chelsea's first-choice left-back last season, as Cole was limited to just 17 Premier League appearances.
Having not been offered a new contract as his old deal expired, Cole opted to look further afield rather than stay in the top
And he said on Tuesday that a move to the Italian capital appealed immediately.
"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," said Cole.
"At that time my agent told me about Rome and how badly they wanted me for their project.
"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 spending two loan spells at Milan - and Cole admitted there is 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 added.
"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 pre-season but I still have a bit to go. Hopefully I'll be fully fit when I come back from [Roma's pre-season tour in] America."
