Both the Blues and Gunners have enjoyed considerable success during the Premier League era and intend to remain in the hunt for top trophies in the present and future.
Arsenal have, however, favoured a more frugal approach to transfer dealings than their neighbours at Stamford Bridge.
That is considered to be holding the club back at present, with Unai Emery being given little backing in the recruitment market.
Arsenal also have continental silverware in their sights, but Cole believes top talent will continue to prefer west over north London if given the choice in upcoming windows.
The former Blues star told BT Sport: “Because the expectation is that they [Chelsea] have always got to be there, which Chelsea have been for the last 20 years, it becomes a crisis quicker than it does at Arsenal.
“With Arsenal, they’re probably not going to win anything, they’re in a 10-year transitional period. Chelsea demand to win things.
“If you’re a player, you’d much rather move to Chelsea than Arsenal, there’s a better quality of player there and you’ve got more chance of winning things.”
While Chelsea continue to compete on multiple fronts this season, Maurizio Sarri has faced questions of his methods during a debut campaign in the Blues dugout.
Some of his selection and tactical decisions have left many confused, with pressure having built around him as a result.
The Italian has also taken to criticising the mindset of his players at times, amid struggles away from home, and needed them to prove a point during a Europa League trip to Malmo on Thursday.
Chelsea were able to deliver for their boss as they edged out Swedish opposition 2-1 in the first leg of a last-32 encounter.
Cole, who is now an academy coach with the Blues, said of that result: “The response was excellent, I think Sarri’s put his marker down, 'I’m going this way, we’re sticking with my philosophy' and the lads bought into it.
“I don’t think there’s much difference between the Chelsea we saw [against Malmo] and the Chelsea of the first two months of the season. Maybe they ran into a very good Bournemouth team and a Manchester City team on immense form, so it inflates things.
“It was a massive result for Sarri. If the lads continue with that kind of intensity then the results could flow, they’ve got a cup final to look forward to in a few weeks, the picture could be very different come the end of the month.”
Chelsea’s next four fixtures will see them face Manchester United in the FA Cup fifth round, Malmo in the second leg of their Europa League tie, Manchester City in the Carabao Cup final and Tottenham in the Premier League.