The Sunday Mirror says the Blues plan to start next season with 32-year-old Cole and his deputy Bertrand on an even footing.
The pair will get equal opportunities to cement their place as the club's first-choice left-back.
Chelsea drew up a development plan for 23-year-old Bertrand last year. They set him a target of making 15 Premier League appearances this season and he has already made 13.
Barring injury or loss of form, Bertrand is then expected to start next term 50-50 with Cole before beginning the 2014-15 season as Chelsea's first-choice left-back.
Chelsea decided against replacing England star Cole because they do not want to block highly-rated Bertrand's path.
It was also calculated that keeping the former Arsenal ace for an extra year on big wages would still be cheaper than trying to sign a replacement.