The Blues launched a late bid for the midfielder, but with Ajax having already sold a number of their star players and with little time left to sign a replacement the Eredivisie side rejected the £40million offer.
Alvarez has confirmed to tried to "force" through the transfer but soon realised it was unlikely to happen and now hopes to join a "big club" soon.
Speaking to ESPN, Alvarez said: "There was interest from Chelsea. Nobody expected that they would make another offer. It was a difficult situation.
"Ajax really couldn't do anything. They didn't have a player in my position. Of course I fought as far as I could. I didn't want to do that too much.
"Because I know the responsibility I have within the club. I am very disappointed, but I am convinced that sooner or later a big club will come for me."