ISIS seized control of Palmyra
, a UNESCO World Heritage Site dating back 2,000 years, last month, prompting fears for the site's survival.
An email sent on behalf of Syria's antiquities chief, Maamoun Abdulkarim, head of the Directorate-General of Antiquities and Museums, said the body had heard four days ago from people in Palmyra of the shrines' destruction.
"ISIS has blown up two ancient Muslim shrines in Palmyra, and has published photos of this awful crime against the Syrian cultural heritage on Facebook," the statement said.
One of the tombs destroyed is that of Mohammed bin Ali, a descendent of Ali bin Abi Taleb, the Prophet Mohammed's cousin, the website DGAM said. It's in a hilly area 4 kilometers (2.5 miles) north of Palmyra.
"ISIS militants also blew up the shrine of Shagaf, known as Abu Behaeddine, a religious figure from Palmyra, dated to 500 years ago. The shrine is located in the oasis 500 meters away from the Ancient City's Arch of Triumph," a statement said.
Images posted on the DGAM website
show dust and debris flying into the air as the shrines are destroyed.
Other monuments, temples and historic buildings have been mined, and a statue of a lion at the entrance to Palmyra's museum has been destroyed, the website reported.
Group: ISIS planted mines, bombs in city
ISIS' capture of Palmyra was followed by the summary executions of scores of captive fighters and residents
, according to the Syrian Observatory for Human Rights, a London-based monitoring group.
The group also reported Tuesday that ISIS had destroyed a shrine in the Palmyra area, but it wasn't clear if it was one of those the government named. ISIS blew it up on the pretext of "removing the landmarks of polytheism," the monitoring group said.
On Sunday, the observatory reported that ISIS fighters had planted homemade bombs and mines around the city.
But it said it wasn't clear whether the explosives were laid with the intention of destroying the city's priceless antiquities or preventing government forces from advancing on Palmyra.
There have been bloody clashes in recent days on the road between the city of Homs and Palmyra between ISIS and government forces backed by loyal militias, according to the group.
Palmyra, also known as the "bride of the desert," is an exquisite collection of ruins in the desert northeast of Damascus that was once a monumental city sitting on an important trade route linking Persia, India and China with the Roman Empire.
Its history as an important caravan city at the crossroads of ancient civilizations is reflected in the eclectic mix of architectural styles found among its colonnades and temples.
British historian and novelist Tom Holland describes the site as "an extraordinary fusion of classical and Iranian influences intermixed with various Arab influence as well."
Destruction of Palmyra wouldn't just be a tragedy for Syria, it would be a loss for the entire world, he told CNN last month.
"This isn't just about Middle Eastern history -- these are the wellsprings of the entire global culture. Mesopotamia, Iraq, Syria, this is the wellspring of global civilization. It really couldn't be higher stakes in terms of conservation."