Cole insists Abraham must rise to challenge the arrival of fellow striker Timo Werner from RB Leipzig represents.
Cole, 38, said: “Tammy Abraham will embrace the challenge of Werner coming in and will improve because his attitude is second to none.
“That's the nature of the beast, if you are at any big club, you know that there is a whole team of people scouring the world looking for someone to come in to take your job, to do your job better than you within the team.
“Players at the top level should know that is normal at a club like Chelsea. If you can't accept that as a reality playing for such a massive club, that's the reason why certain players stay at middle clubs.
“Certainly the lads who have come through the system at Chelsea who I've got to know over the years, their attitude and education that they have been based on, they will embrace the challenge.
“Frank understands the importance of competitive squads to drive you on and push you, I think the players will accept it as part of the reality of playing for Chelsea and it will improve them, I believe in that."