Cole has won every honour in the club game since joining Chelsea. The 32-year-old hopes to give Mourinho strong service in the forthcoming campaign, believing he was below his best under the former Porto, Inter Milan and Real Madrid boss in the Portuguese's first spell in west London.
Cole said: "Having the manager back now from 2006 when he bought me, I still feel I owe him something. I didn't really feel I played as well as I could and as well as I have for Chelsea (since), when he bought me.
"On a personal level, I think I owe him a lot. As a team, I hope we can go on and win many trophies. It's a big club and we need to win trophies.
"With the new players we've got in, we're training hard and we're confident that, hopefully, this season could be a great season for Chelsea again.
"I've been lucky enough to win big trophies. Of course, I hope to win more. Having the manager back, hopefully we can do it all together. As players we owe it to the manager to win the league or the Champions League again."