Last week, the FA's disciplinary committee ordered Terek to play their next home game at a neutral venue and fined them 200,000 roubles ($6,500) for the failure to provide adequate security during a league match against Rubin Kazan on March 17.
Chechnya's strongman leader Ramzan Kadyrov, who serves as Terek's honorary president, insulted FIFA referee Mikhail Vilkov over a loudspeaker, calling him a "donkey" after he sent off Terek captain Rizvan Utsyev late in the game.
"Even if it takes 10 years we're not going to back down," Terek vice-president Khaidar Alkhanov was quoted as saying by local outlet R-Sport on Tuesday after their appeal was rejected by the Russian FA. "We plan to file another appeal with the sport's highest court (CAS) in Lausanne."
Terek had already announced they would host champions Zenit St Petersburg at Anzhi Makhachkala's home arena in neighbouring Northern Caucasus region of Dagestan on Sunday if their appeal was rejected.
The Kremlin-backed Kadyrov said a public fund, named after his late father Akhmat, who was assassinated by a bomb blast at Grozny's old stadium in May 2004, would pay for tickets, meals and transportation to and from Makhachkala.
"Our fans will get a chance to see Terek's next match no matter where it will be played," he said.
Critics accuse Kadyrov of ignoring Russian law and running Chechnya like a fiefdom but President Vladimir Putin has relied upon him to rein in rebels and keep order in the mostly Muslim region, still plagued by militant attacks after two wars against separatists seeking to break from Russia.
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- Ramzan Kadyrov