Mourinho returned to Stamford Bridge - where he won five major domestic trophies between 2004 and 2007 - in June, following six years of success with Inter Milan and Real Madrid.
And 32-year-old full-back Cole, who was brought to the Blues from Arsenal by Mourinho in 2006, has been inspired by the return of the self-styled "Special One".
"I've been lucky enough to win big trophies at Chelsea and of course I hope to win more," Cole told the official Chelsea website.
"Hopefully having the manager back, we can do it all together, and we owe it to him as players to win the league or Champions League again.
"I've won things but from day one when I started playing football, I always wanted to win. That feeling has never gone away and having a manager back now from when [the club] bought me in 2006, I still feel I owe him something.
"I didn't feel I played as well as I could when he bought me, so I owe him a lot, and as a team I hope we can go on and win many trophies," Cole continued.
"It's a big club and we need to win trophies. With the new players, we're training hard and we're confident this season can be a great season for Chelsea again.
"He's always had one aim and one objective, and that's to win. It doesn't matter what you've achieved in football, you have to train well and give 100 per cent for him. That's one thing, he always gives players a chance if they are training well and playing well.
"He's a great manager and proves how good he is with the trophies he has won at different teams. He is great at motivating players and keeping you on your toes, so it's a pleasure to play under him.
"The boys are confident and happy he's come back, and in training you see a new lease of life for some players and hopefully that's only going to benefit the team and the club."