A government push to the south could bring allied Iranian and Russian forces even closer to the increasingly tense frontier with Israel and to US forces based further to the east.
14 fighters were killed in the strike, including Iranian forces. Iran promised strong retaliation for the strike.
Within hours of the attack, a consensus of talking heads and internet commentators concluded that it was Israel that bombed a Syrian military base.
Israel released details on Tuesday about an Iranian "air force" deployed in neighboring Syria, including civilian planes suspected of transferring.
The official, who requested anonymity, would not elaborate.
The April 9 attack was directed exclusively at an arsenal of Iranian weapons, which were stored at the targeted airbase outside Palmyra.
Roni Daniel, military editor for The Israeli News Company, said the disclosure was a signal to Iran that its deployments in Syria "are totally exposed to us, and if you take action against us to avenge (the T-4 strike) these targets will be very severely harmed".
With ISIS largely decimated, Iran's growing footprint in the Middle East has become a major source of concern for Israel, as well as for fellow US ally Saudi Arabia.
Trump must keep forces in Syria to block the route from Tehran to Beirut and to mediate between the Syrian Kurds and Turkey, Doran said.
The Iranian death toll in T-4 was unusually high.
Global media later reported that at least 14 people - including a number of Iranian military personnel - had lost their lives in the raid. The Iranians have also described their cargo flights to Syria as carrying humanitarian aid only.
Earlier this year, Israel shot down an Iranian drone that had been launched from Syria and had flown in Israeli air space.
The remarks came after Israeli Prime Minister Benjamin Netanyahu made several threats against Iran in recent days. While Israel maintained its policy of ambiguity, Russian Federation and the United States both revealed that Jerusalem was behind the devastating strike. Israel argues that Iran has threatened its national security through its support for various, mostly Shiite Muslim militias, especially the Lebanese Hezbollah, which regularly threatens and sometimes clashes with Israel.
This seeming call-to-arms was arguably underscored on Thursday when Iranian Defense Minister Amir Hatami visited Baghdad to promote security ties and to bolster the work of an intelligence-sharing network established there in 2015, with participation from Iraq, Iran, Syria, and Russia.As Newsweek notes, the visit coincided with the unusual move by the Iraqi government to carry out airstrikes against rebel positions in Syria, in a potential sign of closer cooperation with Iran and with the Assad regime over the prosecution of that war.