Manchester United defender Rio said he was shocked by the media reports concerning last Friday's 8-0 World Cup qualifying victory in San Marino. Anton, who has never been capped, is on loan from Queens Park Rangers to Turkish club Bursaspor.
Football Against Racism in Europe contacted FIFA about the complaints and the FA is awaiting official notification of the incident which could lead to England being forced to play behind closed doors.
"While we have no reason to dispute the media reports which are without doubt made for the right reasons of fighting racism, at this time we have not found any recorded evidence of the specific discriminatory chanting referring to Rio and Anton Ferdinand and a vile 'bonfire' song," said Club England managing director Adrian Bevington on Friday.
"We will of course continue to review all of our recorded footage. We recognise the importance of FARE's responsibility to report any incidents to FIFA.
"We will liaise with FIFA and work with them to assist any investigation. Should evidence of any racial chanting be found we would expect action to be taken against individuals," added Bevington in the FA statement.
The FA said it would terminate the England membership of any guilty fans and would expect banning orders to be issued by the courts as a minimum penalty.
"We do not want supporters who chant vile or racist abuse following the England team," added the statement.
The FA, who have worked strenuously to rid England of the hooligan element that followed the team for years, pointed out that "in addition to the officially ticketed fans in San Marino there was also a large number of non-members who did not receive tickets through the FA".
The 34-year-old Rio said on his Twitter account: "You expect and accept banter from fans on the terraces as it's part of what makes the game great but racism is not banter...and from your own fans? WOW.
"Always a small minority who ruin it for others. Let's not jump to conclusions and assume though as it might just have been banter. We'll see after the investigation."
Ferdinand, who won the last of his 81 caps in June 2011, angered some England fans when he withdrew after being recalled to the squad for the match in San Marino and Tuesday's World Cup qualifier against Montenegro that ended in a 1-1 draw.
Coach Roy Hodgson said he did not speak to Ferdinand before naming him in his squad and the defender eventually pulled out, citing a special fitness routine he needs to adopt to help control a long-standing back problem.
After his international withdrawal the experienced player flew to Qatar to work as a TV pundit for the San Marino game.
United manager Alex Ferguson said on Friday that he told Ferdinand to go to London to meet Hodgson face-to-face and explain his decision.
Ferguson added he had no problem with the player's trip to Qatar.
"He trained last week then had Friday, Saturday and Sunday off and he could make his own choice about things," Ferguson told a news conference ahead of runaway leaders United's Premier League match at Sunderland on Saturday.
Asked about the alleged racist songs, Ferguson said: "I think that's modern society I am afraid. We see a bit of that from supporters and the way they react to things".
FIFA is obliged to investigate FARE's complaints and if it decides the songs were racist England could be punished heavily.
Bulgaria and Hungary were forced to play behind closed doors in the last week following earlier cases of racist and anti-Semitic chants at World Cup matches.
It is not unheard of for supporters to turn on their own players and racially abuse them.
Southampton's Poland international goalkeeper Artur Boruc threw a water bottle into a crowd of his own fans this season when they began racially abusing him and said his action cost him a place in the starting lineup for several weeks.