The central city, known as Tadmur in Arabic, is home to a UNESCO World Heritage site and is also a strategic military location in central Syria linked by highways to the cities of Homs and Damascus, some 240 km (150 miles) to the southwest.
"The news at the moment is very bad. There are small groups that managed to enter the city from certain points," Syria's antiquities chief Maamoun Abdulkarim told Reuters.
"There were very fierce clashes."
Abdulkarim, who received UNESCO's Cultural Heritage Rescue Prize last year, said hundreds of statues had been moved to safe locations but called on the Syrian army, opposition and international community to save the site.
"The fear is for the museum and the large monuments that cannot be moved," he said. "This is the entire world's battle."
Islamic State has destroyed antiquities and ancient monuments in neighbouring Iraq and is being targeted by US-led air strikes in both countries.
The Britain-based Syrian Observatory for Human Rights monitoring group said that al-Qaida offshoot Islamic State had captured around a third of Palmyra.
Palmyra's 2,000-year-old monuments, which lie on the south-western fringe of the modern city, were put on UNESCO's World Heritage in danger list in 2013. The ruins were part of a desert oasis that was one of the most significant cultural centres of the ancient world.
The attack is part of a westward advance by Islamic State that is adding to the pressures on President Bashar al-Assad's overstretched military and allied militia that are also losing ground to insurgents in the northwest.
Syrian state television said in a news flash that armed forces had confronted "the Daesh (Islamic State) terrorist group" when it tried to enter a northern Palmyra neighborhood.
A video posted by an activist network on YouTube appeared to show black smoke rising into the sky. The caption dated May 20 said it was footage of air strikes on the city. A communications tower and a citadel could be seen in the video.
The Observatory said the two sides were shelling each other and that the military had carried out air strikes.
Islamic State supporters posted pictures on social media showing what they said were gunmen in the streets of Palmyra, which is the location of one of Syria's biggest weapons depots as well as army bases, an airport and a major prison.