The spin-bowling all-rounder sustained the injury during the Lanka Premier League last month and missed the Asia Cup, where Sri Lanka finished runners-up behind India.
"We are consulting foreign doctors to see whether he needs surgery or not," SLC medical panel chief Arjuna de Silva told the Sunday Times newspaper. "If he does, he will be out for at least three months.
"At the moment, the situation is not that great and it’s highly unlikely that he can play the World Cup."
Hasaranga's absence would be a major blow to the 1996 champions considering the 26-year-old was the leading wicket-taker in the last two T20 World Cups.
De Silva said the SLC were doing everything possible to try to get him fit to play some matches at the very least.
"Since he is a key weapon in our attack, we are looking at other options to see how best we can get his services at least for important games," he said.
"All that depends on the opinions of the consultant to whom we are trying to show his reports."
Fast bowler Dushmantha Chameera also risks missing the World Cup with a pectoral muscle injury.
"Chameera is bowling at the moment but still complains of pain," de Silva said. "We have consulted a doctor in the United Kingdom to see how best we could treat him."
