The 32-year-old England defender penned a contract extension at Stamford Bridge last month after speculation he would be allowed to leave in the summer.
Bertrand, 23, has been the left-back understudy to Cole for several seasons and has forced his way into the Blues' side working in front of Cole in a left midfield role.
He was named by former manager Roberto Di Matteo in such a role in Chelsea's successful Champions League final against Bayern Munich last season and Bertrand is delighted Cole will remain at the club to continue helping him refine his game.
"I have said from the start that he deserves another contract. For me, personally, we have a good relationship. I'm always learning and there is no one better to learn from," he said.
"He has given me plenty of advice over the years. The best advice is not so much what he has given me verbally but just watching him every day and the way he goes about his business."
Bertrand also said his blossoming relationship with Cole helps their understanding when they are partnered on the pitch and he believes he will continue to challenge for the left-back role in the future.
"It (their relationship) reflects in our performances when we play together and I am in front of him at left midfield," he said.
"It is probably the natural progression [to share the left-back role with Cole in the future] but I don't pick the team.
"I just get my head down, train well and hopefully that time will come."
Chelsea midfielder Billy Clifford has extended his loan spell at Colchester until the end of the season after impressing during his initial one-month stay.