Parker's injury is currently being evaluated by Spurs medical staff while the first-team squad are on tour in the United States.
Villas-Boas said, in quotes reported by several national newspapers: "There might be a small tear on his Achilles.
"We are at the moment evaluating the situation regarding a possible operation that can take four to six weeks normally to heal."
Parker's absence could see him sidelined until September, forcing him to miss Spurs' start to the Barclays Premier League campaign.
Parker could also sit out England's friendly against Italy on August 15 and, depending on the operation date and his recovery, the World Cup qualifiers with Moldova on September 7 and Ukraine on September 11.