The 31-year old full-back boasts a wealth of experience at the top level and with his contract entering its final year, Bertrand believes the club would be best served keeping hold of Cole, if only so that he can continue to learn from him at close quarters.
"Yes, I want Ashley to stay," Bertrand told reporters.
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"He has been a fantastic servant for the club and he rightly deserves what he is asking for.
"It doesn’t affect me and my game because I am getting plenty of chances to play.
"I have learnt a lot from him and the longer he stays the more I will continue to learn.
"I probably could have a bit of banter with him about me taking his place in the team because we have a really good relationship but we’re more than happy playing together and to leave it as it is."
Bertrand, 23, was recently rewarded with a new five-year deal at the club after seeing his fortunes change dramatically after Roberto Di Matteo took over the reins.
The youngster was a starter in last season's Champions League final before representing both Team GB and England this summer.