Navas, who played for Real Madrid when they beat Atletico Madrid in the Champions League final on Saturday, has been trying to shake off the injury for around a year, Fedefutbol said in a statement.
He also missed last year's CONCACAF Gold Cup, although he played regularly for Real Madrid this season.
"It is the same injury that Navas has been coping with for the last year and which has not responded to a number of treatments that he has undergone," the statement said.
"Real Madrid will determine the treatment for this case so it can be solved definitively.
"Doctors consider that if it is not treated promptly, the player could suffer a complication in his Achilles tendon."
Navas joined Real Madrid two years ago after his outstanding performances helped Costa Rica reach the 2014 World Cup quarter-finals, beating Uruguay and Italy on the way.
Costa Rica face a tough first-round group against the United States, Colombia and Paraguay.
(Reporting by Brian Homewood, editing by Ed Osmond)
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