"Regime forces and Russian troops are about one kilometre from the city," said Rami Abdel Rahman, head of the Syrian Observatory for Human Rights.
After seizing a string of hilltops overlooking Palmyra, the troops had "the western half of the city" within firing range, Abdel Rahman added.
"They are close to capturing the citadel. IS withdrew from it, but they may have left suicide bombers inside," he warned.
Supported by Russian air strikes and ground troops, Syrian government forces have been battling for weeks through the desert in the central province of Homs to reach Palmyra.
IS jihadists first seized Palmyra in May 2015 and began to systematically destroy the city's monuments and temples, while also looting its many archeological treasures.
They were driven out in March 2016 but recaptured the town last December.
Syrian state media confirmed Wednesday that government forces were now in control of key territory near the city.
"Seizing control of the Mount Hilal and other hilltops overlooking Palmyra is an important step towards the collapse of the terrorist groups in the city," state news agency SANA reported.
A senior military source in Damascus told AFP on Wednesday that the army had also reached a strategic crossroads leading into Palmyra.
"This crossroads is the key to entering the city," the source told AFP.
"Our forces have not yet taken the citadel, but the city is within firing range," he added.
IS has ravaged the city's celebrated heritage, blowing up funerary towers and carrying out mass executions in the city's Roman theatre.
Last month, IS destroyed Palmyra's tetrapylon monument, while satellite images showed damage to the theatre's facade.
The new destruction was condemned by the United Nations as a "war crime."
On Wednesday, two funeral busts damaged by IS after it first captured Palmyra were brought back to Syria after being restored in Italy.
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