PSG had asked the LFP to approve the deal even though the required paperwork arrived after Tuesday's French transfer deadline.
Ligue 1 clubs had to send all contracts to the league by 11:00pm on Tuesday (2200 GMT) before having a further hour to register moves in the FIFA system, which validates all international transfers.
Ziyech had passed a medical and both clubs had reached an agreement for the player to be loaned to PSG until the end of the season.
However, according to two sources close to the deal, the Premier League club failed to send the required documents in time for Ziyech to be registered in the FIFA system which validates all international transfers.
The Netherlands-born Ziyech had been targeted by PSG to add depth to their attack after the departure of Spanish international forward Pablo Sarabia to Wolverhampton Wanderers earlier in January.
The Ligue 1 leaders, who play Bayern Munich in the first leg of their Champions League last-16 tie on February 14, failed to make any new signings in the winter window.
They had also targeted Inter Milan's Slovakia centre-back Milan Skriniar, but he will now stay at the Italian club until the end of the season when his contract expires.
