Russia's FSB security service told President Vladimir Putin that 11 people had been detained following Friday's deadly attack near Moscow, including four people directly involved, Interfax cited the Kremlin as saying on Saturday.

It said FSB security service chief Alexander Bortnikov had reported to President Vladimir Putin that those detained included "four terrorists" and that the service was working to identify their accomplices.

Russia's Investigative Committee said the death toll rose to 93 from the attack in which camouflage-clad gunmen opened fire with automatic weapons at concertgoers near the capital on Friday.

Russian lawmaker Alexander Khinshtein said the attackers had fled in a Renault vehicle that was spotted by police in the Bryansk region, about 340 km (210 miles) southwest of Moscow on Friday night, and disobeyed instructions to stop.

While two were arrested after a car chase, two others fled into a forest, but they were later caught and arrested as confirmed by the Kremlin.

Meanwhile, Interfax quoted the FSB security service as saying that four suspects directly involved in the deadly attack near Moscow on Friday were heading toward Russia's border with Ukraine when they were apprehended early on Saturday.

They had contacts on the Ukrainian side, the security service confirmed, adding that the attack had been carefully planned and the perpetrators had weapons prepared in advance in a cache.

"After committing the terrorist attack, the criminals intended to cross the Russian-Ukrainian border and had relevant contacts on the Ukrainian side of the border," the FSB said, confirming that work is underway to determine all the circumstances of the incident.

On his part, Russian lawmaker and former general Andrei Kartapolov was quoted by Ria Novosti as saying that if Ukraine is found to be behind Friday's attack that killed at least 93 people near Moscow there should be a clear answer on the battlefield.

ISIS earlier claimed responsibility for the attack in a short statement released by Amaq, an ISIS-affiliated news agency, on Telegram on March 22 without giving any evidence.