The Uruguay forward is claimed to have bitten Italy's Giorgio Chiellini and the World Cup organisers are expected to reach a decision on Thursday over a suspension
With Uruguay having progressed to the World Cup second round by beating Italy 1-0, football's governing body began deliberations on Wednesday in the hopes of reaching a quick verdict.
"We have to resolve it either today or tomorrow," Martin Hong, a member of Fifa's disciplinary panel, told reporters on Wednesday. "It's our duty to see justice done."
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The deadline for Suarez and Uruguay to present their defence was at 21:00BST on Wednesday, with time short for a decision to be reached.
Should the forward appeal any ban handed out, Fifa might be forced to take the case to the Court of Arbitration for Sport in order to reach a binding verdict before Uruguay's round-of-16 meeting with Colombia.
A suspension of "fewer than three matches or of up to two months," however, could not be appealed as per Fifa rules.
If Fifa were to judge the incident on the scale of normal red-card offences, a suspension could stretch between three and 24 international matches, but the organisation also has the option of a club and international ban of up to two years.