The Blues were beaten 3-0 at Brighton on Friday night, leaving their hold on fourth place again looking shaky.
Carragher said: "Chelsea has been deemed in the last couple of years a project and I think the natives will be getting restless.
"(Mauricio) Pochettino had an incredible run at the end of the season. I think in terms of the numbers you have to look at, there is not going to be much difference between the seasons.
"I think it is a real worry if they aren't getting Champions League football. They have spent big boy money and they were one of the big boys before they even spent that money."
Carragher also said: "This idea that this project is going to take three years, the problem is that the best players won't wait that long. You can see a frustration starting to build. Cole Palmer is good enough to compete for the biggest honours.
"Nottingham Forest are third and Chelsea have spent over a billion pounds. They are basically playing once a week. They have to get Champions League. If they don't get Champions League it will be a disaster."