The Al-Hail and Arak gas fields, 40 and 25 kilometres respectively from Palmyra, were vital for the regime's generation of electricity for areas under its control, the Syrian Observatory for Human Rights said.
Fierce clashes have rocked Palmyra's outskirts since IS launched an offensive on May 13 to capture the 2,000-year-old world heritage site nicknamed "the pearl of the desert".
Since then, at least 364 people, including combatants from both sides and 62 civilians, have been killed in the battle for the ancient city.
"The military situation is under control in the city, but the clashes are ongoing north and northwest of it," said Talal Barazi, the governor of Homs province where Palmyra is located.
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"At least five civilians, including two children, were killed Sunday night when IS fired rockets on numerous neighbourhoods in Tadmur," the Observatory said, using the Arabic name for the city.
"Everyone is holed up at home," Khalil al-Hariri, head of the local museum, told AFP by phone from Palmyra.
According to Syrian antiquities director Mamoun Abdulkarim, two rockets on Sunday hit the garden of Palmyra's museum which housed statues, sarcophagi and other well-preserved artefacts, without causing any damage.
"The museum was emptied a few weeks ago, and the main objects were transferred to secret safe spaces," he told AFP.
Nevertheless, "there are still pieces that we could not transport because they are fixed on the wall", Abdulkarim added.
The antiquities director, who had called on the international community to prevent a "catastrophe" at Palmyra, said he remained worried about the "ancient city, especially its sculpted sarcophagi".
Fears that IS would destroy the beautiful artefacts spurred urgent calls from UNESCO for international action.