Emery will depart Valencia this summer having secured a Champions League spot by leading the club to third in Spanish table. They finished three points ahead of Malaga despite losing to Real Sociedad in their final outing on Saturday.
They also reached the last four of the Europa League, but lost to tournament winners Atletico Madrid.
"We'll have a new coach next season, that will be Unai Emery," said Fedun.
Emery will be assured of a shot at the Champions League with Spartak as the man he is replacing Valery Karpin coached Spartak to second spot in the Russian Premier League on Sunday to qualify automatically.
"Valery Karpin has built the basis of the team, but we need new coaching solutions," said Fedun.
Karpin is set to set to stay on as a director of football at Spartak.