The video posted by Chechen President Ramzan Kadyrov, a strong Putin ally, portrayed Mr Kasyanov though a sniper's gun sights.
Mr Kadyrov has criticised social network Instagram for removing it.
Speaking to BBC News, Mr Kasyanov said Mr Putin was responsible for Mr Kadyrov's actions.
"He is one of the 45 governors of the [Russian] regions, appointed personally by Mr Putin, and Mr Putin is responsible for his behaviour personally," he told the BBC's Sarah Rainsford.
"He is supposed to stop his undertakings... but unfortunately there's no reaction. The fact that Putin who is guarantor of constitution doesn't stop Kadyrov - means it's some kind of silent encouragement at least."
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Mr Kasyanov, who served as President Putin's prime minister from 2000 to 2004, now leads Parnas, a small liberal opposition party.
He and other Russian opposition politicians have described the sniper video, which appeared after the Chechen leader called Mr Putin's critics "enemies" and "traitors", as a murder threat.
"They want to intimidate us and continue to create pressure on the opposition," Mr Kasyanov said.
"The main purpose is pressure on the opposition. To make fear. So that we stop our active policy."
He described Mr Kadyrov as "one of the most outrageous pieces" of the current Russian political system.
"Like all people in Russia, we should all be worried about what is going on," Mr Kasyanov said.
"But we've chosen our way and we have to stand up and continue to do our job."
The former prime minister argued that a campaign of intimidation was under way because the Kremlin feared the opposition speaking out about the economic crisis and other issues.
"That's why they are afraid of us," he said. "Because we will describe  who is responsible for the problems of the people."
Instagram said the controversial video had "violated the requirement to respect other members" of the social network.
Mr Kadyrov, who has ruled Chechnya with an iron fist for nearly a decade with the support of the Kremlin, suppressing separatism, accused Instagram of bowing to US pressure.
In a new Instagram post, he said he had been punished for saying a "few words about the USA's guard dogs".
"You can write anything you want, but do not touch America's dogs, friends of the Department of State and Congress," he said. "You know very well whom I am talking about!"
Last March Mr Kadyrov spoke out on Instagram about the assassination of opposition figure Boris Nemtsov in Moscow, defending one of the Chechens charged over the shooting.
A day later President Putin gave Mr Kadyrov a top award.
Nemtsov, also a leading member of Parnas, was among several well-known opponents of Mr Putin assassinated in the past decade.
Russian politician Mikhail Kasyanov has accused President Vladimir Putin of "silently encouraging" intimidation as a row over a menacing video continues.