There had been speculation Moscow could boycott the Games entirely after the International Olympic Committee (IOC) on Tuesday barred the country from competition over what its chief Thomas Bach described as Russia's "unprecedented attack on the integrity of the Olympic Games and sport".
Clean Russian athletes would be allowed to compete under an Olympic flag, the IOC said.
"We will certainly not declare any boycott, we will not prevent our Olympians from competing, if they want to take part in a personal capacity," Putin said after a speech at an automobile plant in Nizhny Novgorod, in which he announced he would run for a fourth presidential term next year.
"The final decision of course must be made by the Olympic team."
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Putin said the ban "looks like an absolutely staged and politically motivated decision".
"We all see this, for me there is no doubt about it," he stated, repeating denials that any such state doping programme had existed in Russia.
Putin's spokesman Dmitry Peskov earlier Wednesday cautioned against an "emotional" response to the ban.
"The situation is serious, it calls for deep analysis," he said.
Russia has always denied any state doping programme, and politicians and athletes earlier reacted with anger and disappointment to the IOC decision.
The head of Russia's Olympic Committee, Alexander Zhukov, told the IOC that punishing clean athletes was "unjust and immoral".
Russian media expressed regret at the decision while welcoming the possibility of some athletes participating in South Korea next year, albeit under tight restrictions.
The IOC "chose one of the harshest options it was considering but still not the harshest of all", which would have been a total ban, wrote Kommersant business daily.
It slammed the decision as "very harsh and in some ways even humiliating for Russia," citing the life bans on attending the games for ex-sports minister Vitaly Mutko, now first deputy prime minister.
"Can't get by without Russia," the pro-Kremlin Izvestia daily headlined its front page, stressing that "Russian Olympic athletes will defend the honour of the Motherland under any banner."
Some top sports figures agreed, with ice hockey forward Ilya Kovalchuk telling TASS state news agency: "We must go to the Olympics. Refusing is surrender."
- 'This is sport, damn it!' -
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Some politicians vented rage, however.
"This is an absolute outrage," said the last Soviet leader Mikhail Gorbachev, quoted by state RIA Novosti news agency. "This is sport, damn it!" he added.
Nationalist politician Vladimir Zhirinovsky, leader of the Liberal Democratic Party, condemned the decision as "sporting racism, political racism."
Chechen strongman Ramzan Kadyrov, writing on his Telegram social networking page, warned that "not one athlete living in Chechnya will participate under a neutral flag."
Pro-Kremlin media focused on discrediting Grigory Rodchenkov, the whistleblower who gave evidence of a state- controlled doping programme in which he played a central role.
Rodchenkov has been living in hiding in the United States since lifting the lid on the intricate workings of a state- supported doping scheme at the 2014 Winter Olympics in Sochi.
"Grigory Rodchenkov is the perfect traitor," wrote tabloid daily Komsomolskaya Pravda.