The jihadists group first took the city in May 2015 until they were ousted by the government forces in March this year.
Russian Federation has carried out a bombing campaign in Syria in support of its ally Syrian President Bashar al-Assad since September a year ago.
Peskov said numerous IS fighters involved in the Palmyra assault had recently left the Iraqi city of Mosul, where US -backed Iraqi forces have been waging a almost two-month-old offensive against the militants.
The Russian Defense Ministry claims that its jets launched 64 airstrikes, killing more than 300 militants around Palmyra, according to state-controlled media.
The collapse of the city's defenses despite the heavy bombing and reinforcements sent by the Syrian army has exposed the limitations of the Russian backing that has turned the tide of the conflict in President Bashar al-Assad's favor.
Russian Federation said on Sunday Islamic State had deployed over 4,000 fighters in the assault drawing on significant forces from their strongholds in Raqqa and Deir Zor.
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"After capturing Tadmur (Palmyra) the Khilafah soldiers continued their progress today after ferocious clashes with the army and its militias", the statement by Islamic State's Homs Wilayat (province) said.
Tens of thousands of civilians have fled to government-held western Aleppo since the offensive began earlier this month, but thousands remain trapped in the rebel stronghold.
In May a year ago, the Sunni Muslim extremist group seized several towns in Homs province including Palmyra, where they caused extensive damage to many of its ancient sites.
In the centuries that followed, the area "stood at the crossroads of several civilizations", with its art and architecture mixing Greek, Roman and Persian influences, according to UNESCO, the United Nations agency that documents the world's most important cultural and natural sites. But hours later, the activists said IS had seized a castle just outside the town that overlooks its famed Roman-era ruins.