"Israel is determined to continue to prevent the transfer of advanced weapons to Hezbollah," the senior official told The New York Times.
"The transfer of such weapons to Hezbollah will destabilize and endanger the entire region.
"If Syrian President Assad reacts by attacking Israel, or tries to strike Israel through his terrorist proxies, he will risk forfeiting his regime, for Israel will retaliate."
The official, who the Times said had been briefed by high-level officials on Israel's assessment of the situation in Syria, contacted the newspaper Wednesday.
The reason for Israel's warning was not immediately clear.
Analysts suggested to the Times Jerusalem could be seeking to restrain Damascus to avoid taking further military action, or to alert other countries to a possible additional military strike.
The analysts said Israel could also be warning Hezbollah and supporter Iran, the Times said. Hezbollah, a Shiite Islamic militant group and political party based in Lebanon, recently said it could use weapons supplied by Iran to retaliate for Israeli strikes on Syria early this month.
Hezbollah also gets financial and political support from Syria.
Israel, the United States and six other governments classify Hezbollah as a terrorist organization, in whole or in part.
Israeli warplanes carried out two strikes in Syria May 3 and 5. The first, at Damascus International Airport, is reported to have sought to destroy weapons being sent to Hezbollah from Iran. The second is reported to have hit Syrian Republican Guard bases and long-range-missile storehouses, along with a military research center U.S. officials say was the country's main chemical weapons site.
Jerusalem did not confirm either attack, which followed another one Jan. 31 that was reported to have hit a convoy carrying advanced antiaircraft weapons to the Hezbollah militia.
A spokesman for Israeli Prime Minister Binyamin Netanyahu declined to comment on the Times report.
Damascus had no immediate comment on the report.
But after the early May attacks, it condemned Israel, saying the assaults "opened the door to all possibilities."
"We will respond immediately and harshly to any additional attack by Israel," Deputy Foreign Minister Faisal Mekdad said.
The Israeli official told the Times Wednesday, "Israel has so far refrained from intervening in Syria's civil war and will maintain this policy as long as Assad refrains from attacking Israel directly or indirectly."
He added, "Israel will continue its policy of interdicting attempts to strengthen Hezbollah."
White House spokesman Jay Carney also declined to comment on the report.
But "broadly speaking, we have long acknowledged and recognized that it is part of Israel's sovereign right to defend itself and that its concern about the transfer of sophisticated weaponry to Hezbollah is legitimate," Carney said.
"We understand that concern and why Israel holds it, and why they take action to address it," he said. "Again, that's not in response to any specific reported action but simply our position on the overall matter."