Hezbollah did not immediately comment after Sunday's claims.
Iran said it will stand by Syria, "and if there is need for training, we will provide them with necessary training," Brig. Gen. Ahmad-Reza Pourdastan, commander of the Iranian Army's Ground Forces, told reporters Sunday.
Iranian Foreign Ministry spokesman Ramin Mehmanparast said he had no doubt Syria and its allies will "give a crushing response to the aggressions of the Zionists," the state-run IRNA news agency reported.
Russia also weighed in Monday, with a Foreign Ministry spokesman calling the reports of Israeli strikes "very worrying."
"Any intensification of military confrontation greatly increases the risks of creating hotbeds of tension aside from Syria, in Lebanon, and also destabilizing the Israeli-Lebanese border, which has so far remained relatively calm," ministry spokesman Alexander Lukashevich said Monday.
But Israeli Maj. Gen. Yair Golan indicated war is not imminent, according to the IDF's website.
"There are no winds of war," said Golan, who is in charge of the Northern Command.
Two rockets fired from inside Syria fell into the Golan Heights, the IDF said in a tweet. The military said the rockets were "fired erroneously as a byproduct of internal conflict in Syria."
NATO Secretary General Anders Fogh Rasmussen told reporters Monday that the alliance had no information on the reported airstrikes, but said the alliance remains concerned about the possibility that the conflict could spread beyond Syria's borders.
"It's not a new concern that for quite some time we have expressed concerned of the risk of spillover of this conflict," he said.
Chemical weapons reports
The tensions have been worsened by conflicting reports on the possible use of chemical weapons in Syria.
On Monday, a U.N. official said evidence points to the use of the deadly nerve agent sarin by Syrian rebel forces.
Carla Del Ponte told an Italian-Swiss TV station that the findings come after interviews with doctors and Syrian victims now in neighboring countries.
Del Ponte, the commissioner of the U.N. Independent International Commission of Inquiry for Syria, said the notion isn't surprising, given the infiltration of foreign fighters into the Syrian opposition.
Later, the commission issued a press release saying it "has not reached conclusive findings as to the use of chemical weapons in Syria by any parties to the conflict."
Therefore, "the commission is not in a position to further comment on the allegations at this time," the statement said.
The Syrian Coalition released a statement in response to Del Ponte's comments, condemning the use of chemical weapons and stressing that it will continue to conduct its own investigations.
"The Syrian Coalition will take all necessary legal measures in case the investigation reveals parties other than Assad's regime have used chemical weapons," it said.
A U.S. State Department official told CNN that the United States does not have information suggesting that rebels have "either the capability or the intent to deploy or use such weapons."
But, the source said, the "facts are not complete" and efforts to obtain more information are ongoing.
Rebel Free Syrian Army spokesman Louay Almokdad said rebels don't have unconventional weapons, nor do they want any.
"In any case, we don't have the mechanism to launch these kinds of weapons, which would need missiles that can carry chemical warheads, and we in the FSA do not possess these kind of capabilities," Almokdad said.
"More importantly, we do not aspire to have (chemical weapons) because we view our battle with the regime as a battle for the establishment of a free democratic state. ... We want to build a free democratic state that recognizes and abides by all international accords and agreements -- and chemical and biological warfare is something forbidden legally and internationally."