The former Blues midfielder was speaking after Arsenal suffered a shock 1-0 Europa League away defeat to BATE Borisov, while Chelsea recovered from their 6-0 Premier League capitulation against Manchester City by beating Malmo 2-1 in Sweden.
The two clubs brought in new managers last summer, Maurizio Sarri and Unai Emery respectively, and both have come in for criticism this season.
However, Cole insists that Stamford Bridge boss Sarri is always going to face more scrutiny as he is expected to deliver silverware - unlike Emery.
“Everyone has settled into Arsenal over the last 10-12 years, that it’s ok for Arsenal not to win things,” he told BT Sport.
“But with Chelsea when they don’t win – a bit like how Manchester United were when things went bad for them – because the expectations are always there, and they have been with Chelsea over the last 20 years; when they go on a bad run like this it becomes a crisis quicker than it does at Arsenal.
“With Arsenal, people just accept that they’re probably not going to win anything and they’re in this 10-year transitional period.
“But with Chelsea the pressure is on, the club demands to win things, so I think everyone looks at it in a different way.
“If you’re a player and you’re going to move to Chelsea or Arsenal now, you’re going to prefer Chelsea because there’s a better quality of players there, a better standard, and you’ll have more chance of winning things.”
