Former United States attorney Garcia carried out a report into the bidding processes behind the 2018 and 2022 World Cups in Russia and Qatar respectively, amid claims of wrongdoing, and a summary of his findings was released last week.
Yet Garcia, who led the investigation by FIFA's Ethics Committee, was critical of the way his report was presented, stating there were "numerous materially incomplete and erroneous representations of the facts".
FIFA has since received Garcia's intention to appeal the report, after their overall findings into the inquiry claimed "the various incidents which might have occurred are not suited to compromise the integrity of the FIFA World Cup 2018/2022 bidding process as a whole".
Dyke, writing to members of the Executive Committee on behalf of the FA, called on the ExCo to insist on the publication of the full report.
"As you probably know the reputation of FIFA was already low in England and much of Europe before the events of last week," Dyke wrote.
"The failure to publish Mr Garcia's report and his statement that the summary report which was published contained "numerous materially incomplete and erroneous representations" has resulted in a further decline in public confidence of FIFA.
"We cannot go on like this. Complete transparency is required if the actions of all those who bid, including England 2018, are to be judged fairly.
"I know some of you believe that FIFA’s reputation in England is the result of an obsession amongst the English media with FIFA and I know Mr [Sepp] Blatter sees their reports as an unfair attack on the organisation he leads.
"However, in England we see it differently. If you read a whole range of English newspaper reports about FIFA, particularly those in the Sunday Times, they do provide compelling evidence of wrong doing.
"They cannot be simply dismissed as "racist" or "an attack on FIFA" as Mr Blatter described them at the FIFA Congress in Brazil.
"Urgent action is needed if confidence in FIFA is to be rebuilt in England. The FA is of the view that this action should start with the full publication of Mr Garcia's report."
