B. Mourad, 34, lives in the French department of Savoie and says he is a former agent of the Algerian secret services. He told AFP Friday that he may face the death penalty in Algeria.
Mourad says he left Algeria in 2011 and several times asked for asylum in France.
On 12 August the French Office for the Protection of Refugees and Stateless People (OFPRA) rejected his request for reconsideration of the case, according to a document consulted by AFP.
In its decision, the OFPRA argued that "the person concerned (...) has not been able to produce a compelling explanation to establish the truth" of his collaboration with the Algerian intelligence services.
"If I am sent back to Algeria, I will be tried by the military court for treason and could get the death penalty," Mourad told AFP.
It is just the latest chapter in the mystery surrounding the beheadings of the seven monks. Their deaths have long been a source of controversy in France and Algeria.
Their deaths were initially claimed by the Armed Islamic Group of Algeria. But a retired French general claimed in 2009 that the monks had been accidentally shot dead by troops in an Algerian military helicopter during a botched rescue attempt. The general said the heads were removed afterwards.
In a report made public last month, a team of three experts concluded the monks were killed several weeks before the date claimed by the Armed Islamic Group of 21 May 1996.
That enquiry, after a long investigation overseen by a French anti-terror judge, has shed doubt on whether the deaths were in fact the work of jihadist rebels.