The governing body issued a statement after comments on social networking site Twitter appeared following England's Euro 2012 defeat to Italy in which the pair missed penalties
After initially taking a lead in the shoot-out, the Manchester United winger hit the crossbar, before the Chelsea left-back's effort was saved by Gianluigi Buffon, allowing Alessandro Diamanti to slot the winning penalty into the back of the net.
The FA issued a statement, after it was alleged that the English duo suffered racial abuse on Twitter following the clash, stating their disapproval of the reported comment and that they fully back any police action.
The statement read: "We are concerned at the reports regarding allegations of abuse aimed at England players Ashley Cole and Ashley Young on Twitter.
"They have just given everything for the national team at Euro 2012 and it is appalling and unacceptable that messages of an abusive type are being posted.
"We support any police investigation in identifying who is behind this."