Terry has confirmed he will be leaving Stamford Bridge on a free transfer at the end of the season, ending his 22-year stay as a Chelsea player.
The former England captain has been studying for coaching badges while also learning Portuguese in preparation for the next stage of his career, but insists he wants to continue playing despite a lack of game time this season.
Chelsea lead the Premier League title race and are looking forward to an FA Cup Final against Arsenal on May 27 without Terry as a regular starter, while coach Conte preferred to start youngster Nathan Ake in the Blues' semi-final win against Tottenham.
Although Terry has slipped down the pecking order at Chelsea, former Stamford Bridge winger Cole feels that Conte's squad will miss the 36-year-old's "remarkable" presence in the dressing room.
Cole - now plying his trade in America for the Tampa Bay Rowdies - told the Daily Mail: "When I signed for Chelsea [in 2003], he was the first person to see me as I walked through the door.
"Although he's only a few months older, he put his arm around me and it was like, 'I'll look after you' like the new kid at school, like an older brother.
"Chelsea will miss that. They are a well-run club but forget everything he does on the pitch, it's the unseen things as a captain.
"I remember I was having a bit of a rough time there. He recognised it and got one of the video analysts to make a compilation of all my best bits. It was a little touch but it really lifted me.
"For him to even notice that... in football you're so focused on yourself and your own game, to have an eye on everyone else in the club is remarkable.
"You can't quantify that. He was a younger man then, as well, and to have that empathy - that's one of a million things the club will miss.
"They'll have to spend a lot of money to replace him. He'd love it out here. He'd be perfect for any club out here. Anyone would take him."