World football's governing body initially imposed the punishment in January due to the capital clubs breaching rules in relation to the transfer of minors but suspended them later in the month, pending appeal.
Madrid and Atletico will now be unable to register new players until January 2018, with their respective initial fines of 360,000 Swiss francs and 900,000 Swiss francs remaining.
Barcelona were given a similar punishment in 2014 and signed Arda Turan and Aleix Vidal before the start of last season, even though the registration ban prevented either man from representing the club until January this year.
In a statement Real Madrid confirmed their intention take their case against the sanctions to the Court of Arbitration for Sport (CAS).
Their statement read: "Faced with the decision the Appeal Committee of FIFA has released today, Real Madrid CF states the following.
"1. We regret the decision on the ground that [it] is profoundly unfair and contrary to the most elementary principles of penal law.
"2. The club will make a corresponding appeal to the Court of Arbitration for Sport, requesting full revocation of the same, with the full conviction of obtaining a fully favourable decision before that body.
"3. Real Madrid CF will request that the procedure is processed by way of urgency in order to obtain the termination of [it by] CAS as soon as possible."
Barcelona's appeal also allowed them an extra transfer window in which to operate before the suspension came into effect, during which time they signed Luis Suarez, but Madrid were uncharacteristically quiet over the recent close-season.
A report by the Daily Mirror earlier on Thursday suggested current Santiago Bernabeu stars Cristiano Ronaldo, Gareth Bale, Toni Kroos and Luka Modric were in line for lucrative contract renewals.
Atletico also confirmed their intention to take an appeal to CAS.
In a statement, they said: "Our organisation wishes to state that one hundred per cent of the licences of our players who are part of this investigation and continue at the club have been regularised by FIFA itself, receiving authorisation from the sub-committee to process our licences and compete in different categories."
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