Both players missed penalties in the Euro 2012 quarter-final shoot-out defeat against Italy in Kiev and it has since transpired that an attack was launched by one user on Twitter.
"We are concerned at the reports regarding allegations of abuse aimed at England players Ashley Cole and Ashley Young on Twitter," an FA statement read.
"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."
Cole and Young, who returned home with the remainder of the team on Monday afternoon, were singled out for abuse on Twitter by one user following the 4-2 shoot-out defeat.
In a series of offensive postings, the user goaded others who criticised him for his rants.
In one post the tweeter wrote: "All I got to say is this was just a joke, you guys need to relax, police know i was kiddin around and they won't bother me hopefully, cheers."
Another said: "The police won't bother me mate i can assure you that."
A Metropolitan Police spokesman said: "We are aware of alleged racist comments on Twitter following last night's England game and have launched an investigation.
"The allegation was made to MPS on 25 June by a member of the public relating to comments on a Twitter account supposedly based in London.
"No arrests at this early stage.
"The MPS is examining possible offences under Section 18/19 Public Order Act 1986."
Student Liam Stacey, 21, from Pontypridd, South Wales, was jailed for 56 days in March after racially abusing two Twitter users and mocking Bolton midfielder Fabrice Muamba after he collapsed and almost died.
Stacey, who was in the final semester of his three-year biology degree when he drunkenly sent his offensive and now infamous tweets, was also banned from Swansea University for the rest of the year.
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