The 26-year-old has been the subject of three bids from Italian club Roma since expressing his desire to leave the 2015-16 Premier League champions.
The FAF said he had been given permission to miss training to agree a deal before the transfer deadline.
Barcelona, Manchester United and Chelsea are also said to be interested.
But Arsenal are not in contention to sign the winger.
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Roma’s third bid was reported to be £32m and the club have said they are now looking at other options.
A statement from the FAF said Mahrez “was authorised by the national coach Lucas Alcaraz and the Algerian Football Federation to make an express trip to Europe to formalise on Thursday his transfer to his new club.
The national team will fly without him on Thursday morning to Lusaka to face Zambia on Saturday.”
Mahrez was named the PFA Player of the Year after playing a key part in Leicester’s title-winning season.
He has featured in all three of the Foxes’ Premier League fixtures so far this term.
