The Swiss FA was given a deadline of January 13 to follow FIFA's instructions or face a ban which would also result in FC Basel being expelled from the Champions League.
"Should this deadline not be respected, the Swiss FA (SFV) will be automatically suspended from 14 January 2012 onwards," the statement said.
If suspended, Switzerland - due to host Argentina in a friendly in February - would not be able to play any international matches and FC Basel would be unable to take part in the Champions League round of 16 tie against Bayern Munich.
Euro 2012 would not be affected, however, as Switzerland did not qualify for the finals.
FIFA have demanded that Sion be penalised with a 3-0 defeat for every match in which they fielded any of six players signed during the summer when the club was subject to a transfer ban.
The warning follows a complex legal battle in which Sion defied FIFA and UEFA statutes by taking its case to a civil court.
"The executive committee decided to give a final deadline of January 13 to the Swiss FA to enforce the registration ban imposed on Sion... and to sanction the attitude of the club repeatedly trying to circumvent this decision in a legally abusive manner," said FIFA.
"As a consequence, all matches in which the relevant players participated shall be declared forfeit or three points shall be deducted respectively."
DEMAND 'IMPRACTICAL'
SFV President Peter Gilleron told a news conference FIFA's demand was "impractical" although he believed a points deduction would be possible.
"There are many complex juridical matters to clear up," he said.
The case began when Sion signed six players despite being banned from doing so after FIFA found them guilty of poaching a player from an Egyptian club in 2008.
The six players took their case to a civil court in the canton of Valais, which ruled they could play, and Sion subsequently fielded them in the Swiss league.
Some of the players also appeared in a Europa League qualifying tie against Celtic. Sion won the tie but were expelled from the competition by UEFA who ruled the players were ineligible.
Sion then obtained an injunction at another court in the canton of Vaud, where UEFA's headquarters are based, ordering European football's governing body to reinstate them to the competition.
However, this injunction referred the matter to the Court of Arbitration for Sport (CAS) which on Thursday ruled in favour of UEFA.
The injunction obtained by the players in Valais canton was also overturned by a higher court.
The case has already caused acute embarrassment to Switzerland with other clubs admitting it has damaged the country's image in the sport.
At one point, UEFA President Michel Platini appeared before prosecutors in Vaud to explain why his organisation had not complied with the injunction ordering Sion's reinstatement to the Europa League.
