The 33-year-old defender is now a free agent after leaving Chelsea last month, with talk that his next move will either be to Serie A or the MLS, where he would be following in Frank Lampard's footsteps.
Lampard is set to move to the United States to play for New York City next March, and could feature for Melbourne City on a short-term deal in the interim.
Reports have also linked Cole - who made more than 200 appearances over eight years at Stamford Bridge - with a move to Roma.
Chelsea bid a fond farewell to their defensive stalwart in a statement which read: "We say goodbye to Ashley Cole, a player who contributed enormously to the highly successfully period in our history in which we were crowned champions of Europe and won the League and Cup double for the first time.
"His remarkably consistent high level of performance was always appreciated by the Chelsea support.
"He collected winners' medals for the Champions League, the Premier League, the Europa League, the League Cup, the Community Shield and the FA Cup four times, taking him to his all-time record seven wins in that competition's long history."