The veteran left-back will miss around two weeks after hobbling off in the FA Cup quarter-final replay win over Manchester United on Monday and Bertrand hopes he is the man to replace the England international.
"It's down to me to keep training well, keep my performances up to a level and keep pushing and pushing until the time comes," Bertrand told the club's official website ahead of Thursday's Europa League clash with Rubin Kazan.
"He (Cole) has been a focal point for my career even before he was at Chelsea.
"He was someone I used to watch when I was younger and having him here at Chelsea has been a good chance for me to see first-hand how he goes about his work day in, day out and he provided a good basis for me to plan around.
"Personally it's a chance for me to come in now and play in the next few games, but it's just important to focus on the next game and make sure we get the right results."