Gilleron said that FIFA was asking the SFV to break its own statutes by instructing it on how to deal with first division club Sion, who signed six players in the summer and then fielded them despite being subject to a transfer ban.
FIFA's Executive Committee, meeting in Tokyo on Saturday, said that Sion must be punished with 3-0 defeats for every match in which any of the players were fielded.
"It was a total surprise," Gilleron told the SFV's website.
"We knew that the Sion case would be discussed at FIFA's meeting in Tokyo and we were prepared for all eventualities.
"This decision was naturally a disappointment for us."
"I cannot understand the decision, our arguments were not sufficiently taken into considering. Absolutely not."
"We have no influence over the disciplinary commission," he said. "It's independent, and we place great value on this division of power."
"The decision, to punish Sion with forfeits, could also be a case of the Court of Arbitration for Sport," he added
"I'm sure we will find a solution, although the deadline is tight."
European football's governing body UEFA said in a statement that it had not decided what to do if Basel, who qualified at the expense of Manchester United, could not play in the last sixteen.
"UEFA is confident that the SFV will comply with the FIFA request before the deadline of 13th January and before the UEFA Champions League round of 16 first leg on 22nd February," it said.
"Should this not be the case UEFA will review the situation nearer the time."
