Alex spent a total of eight years on the Blues' books after joining from Santos in 2004, although he was loaned out to PSV for three seasons after work permit issues delayed his Premier League introduction.
The Brazilian helped Chelsea lift the Premier League title in 2009-10 and formed part of the team that reached the Champions League semis the previous season, before being sold to Paris Saint-Germain in January 2012.
But he has now affirmed that he could have moved to the Gunners after becoming frustrated by a lack of first-team football in west London.
"We talked when I was about to leave Chelsea. I was away, I received a call from an Arsenal director, who asked me the values of my salary, if I was interested," Alex explained to Goal.
"The conversation was very quick, but he did not call again. In the end, I sealed a move to PSG.
"My wish was not to leave Chelsea, but I had been training alone for three months, it was not easy.
"At first me and [Nicolas] Anelka trained together, but then he left, and I was alone. If Arsenal had come up with a good, concrete proposal, I would have taken it. I repeat: it was not a desire to trade Chelsea for Arsenal, but I would have accepted it. I wanted to play."
Alex has not played professional football in two years, having been released from last club Milan at the end of the 2015-16 season.
The centre-back then announced his retirement from the game due to a serious knee injury - but he is now reconsidering that call.
"My knee? Thank God I'm healing well, feeling better every day. Of course, it is not easy, it takes training and doing gym work every day to stay strong," he added.
"I'm still not 100 per cent, I'm recovering every day. It is a cartilage injury, so you have to take a little more care. It is good to avoid overdoing it, the impact may not do the knee any good the next day. But I'm managing it well.
"At the end of 2016, my wife and I sat down, and as we talked to the doctors, we felt it would be very difficult to continue playing at a professional level, so we decided to announce that I would stop. Today I do not see it this way, because of the training I've been doing and my recovery.
"It took a little longer than expected, but I felt better. I've been training more in the sand, but I'm better and more confident. If I feel even better and there is some possibility I would like [to return to play]. Stopping playing is the most difficult decision in a player's life."